\declaretheorem

[name=Theorem,numberwithin=section]thm

Privacy-Preserving Public Information for Sequential Games

Avrim Blum Blum, Morgenstern, and Sharma were partially supported by NSF grants CCF-1116892 and CCF-1101215. Morgenstern was partially supported by an NSF GRFP award and a Simons Award for Graduate Students in Theoretical Computer Science. Computer Science Department
Carnegie Mellon University
Jamie Morgenstern Computer Science Department
Carnegie Mellon University
Ankit Sharma Computer Science Department
Carnegie Mellon University
Adam Smith Computer Science Department
Penn State University

Abstract

In settings with incomplete information, players can find it difficult to coordinate to find states with good social welfare. For instance, one of the main reasons behind the recent financial crisis was found to be the lack of market transparency, which made it difficult for financial firms to accurately measure the risks and returns of their investments. Although regulators may have access to firms’ investment decisions, directly reporting all firms’ actions raises confidentiality concerns for both individuals and institutions. The natural question, therefore, is whether it is possible for the regulatory agencies to publish some information that, on one hand, helps the financial firms understand the risks of their investments better, and, at the same time, preserves the privacy of their investment decisions. More generally, when can the publication of privacy-preserving information about the state of the game improve overall outcomes such as social welfare?

In this paper, we explore this question in a sequential resource-sharing game where the value gained by a player on choosing a resource depends on the number of other players who have chosen that resource in the past. Without any knowledge of the actions of the past players, the social welfare attained in this game can be arbitrarily bad. We show, however, that it is possible for the players to achieve good social welfare with the help of privacy-preserving, publicly-announced information. We model the behavior of players in this imperfect information setting in two ways – greedy and undominated strategic behaviours, and we prove guarantees on social welfare that certain kinds of privacy-preserving information can help attain. To achieve the social welfare guarantees, we design a counter with improved privacy guarantees under continual observation. In addition to the resource-sharing game, we study the main question for other games including sequential versions of the cut, machine-scheduling and cost-sharing games, and games where the value attained by a player on a particular action is not only a function of the actions of the past players but also of the actions of the future players.

1 Introduction

Multi-agent settings that are non-transparent (where players cannot see the current state of the system) have the potential to lead to disastrous outcomes. For example, in examining causes of the recent financial crisis and subsequent recession, the Financial Crisis Inquiry Commission [5, p. 352] concluded that “The OTC derivatives market’s lack of transparency and of effective price discovery exacerbated the collateral disputes of AIG and Goldman Sachs and similar disputes between other derivatives counterparties.” Even though regulators have access to detailed confidential information about financial institutions and (indirectly) individuals, current statistics and indices are based only on public data, since disclosures based on confidential information are restricted. However, forecasts based on confidential data can be much more accurate¹¹1For example, Oet et al. [14] compared an index based on both public and confidential data with an analogous index based only on publicly available data. The former index would have been a significantly more accurate predictor of financial stress during the recent financial crisis (see Oet et al. [13, Figure 4]). See Flood et al. [6] for further discussion., prompting regulators to ask whether aggregate statistics can be economically useful while also providing rigorous privacy guarantees [6].

In this work, we show that such privacy-preserving public information, in an interesting class of sequential decision-making games, can achieve (nearly) the best of both worlds. In particular, the goal is to produce information about actions taken by previous agents that can be posted publicly, preserves all agents’ (differential) privacy, and can significantly improve worst-case social-welfare. While our models do not directly speak to the highly complex issues involved in real-world financial decision-making, they do indicate that in settings involving contention for resources and first-mover advantages, privacy-preserving public information can be a significant help in improving social welfare. In the following sections, we describe the game setting and the information model.

1.1 Game Model

Consider a setting in which there are $m$ resources and $n$ players. The players arrive online, in an adversarial order, one at a time²²2For ease of exposition, we rename players such that player $i$ is the $i$ th to arrive.. Each player $i$ has some set $A_{i}$ of resources she is interested in and that is known only to herself. An action $a_{i}$ of player $i$ is of the form $(a_{i,1},\ldots,a_{i,m})$ , where $a_{i,r}\geq 0$ represents the amount that player $i$ invests in resource $r$ , and moreover, $\sum_{j\in[m]}a_{i,j}=1$ . For simplicity, we assume that all $a_{i,r}$ are in $\{0,1\}$ i.e, the unit-demand setting (we study the continuous version where $a_{i,r}$ ’s can be fractional, but still sum to 1, in Appendix C). Furthermore, we do not make the assumption that players have knowledge of their position in the sequence, that is, a player need not know how many players have acted before her.

Each resource $r$ has some non-increasing function $V_{r}:\mathbb{Z}^{+}\rightarrow\mathbb{R}^{+}$ indicating the value, or utility, of this resource to the $k$ th player who chooses it. Therefore, the utility of player $i$ is $u_{i}(a_{i},a_{1,\ldots,i-1})=\sum_{r}a_{i,r}V_{r}(x_{i,r})$ , where $x_{i,r}=\sum_{j=1}^{i-1}a_{j,r}$ for each $r$ . In this resource sharing setting, the utility for a player of choosing a certain resource is a function of the resource and (importantly) the number of players who have invested in the resource before her (and not after her)³³3In Section 5, we consider a generalization where the utility to a player of investing in a particular resource is a function of the total number of players who have chosen that resource, including those who have invested after her..

Illustrative Example

For each resource, suppose $V_{r}(k)=V_{r}(0)/k$ , where $V_{r}(0)$ is the initial value of resource $r$ . The value of each resource $r$ drops rapidly as a function of the number of players who have chosen it so far. If each player $i$ has perfect information about the investment choices made by the players before her, the optimal action for player $i$ is to greedily select the action in $A_{i}$ of highest utility based on the number of players who have selected each resource so far. As shown in Section 3, the resulting social welfare of this behavior is within a factor of 4 of the optimal. In the case where each player has no information about other players’ behaviors, some particularly disastrous sequences of actions might reasonably occur, leading to very low social welfare. For example, if each player $i$ has access to a public resource $r$ where $V_{r}(0)=1$ and a private resource $r_{i}$ where $V_{r_{i}}(0)=1-\epsilon$ , each might reasonably choose greedily according to $V_{\cdot}(0)$ , selecting the resource of highest initial value (in this case, $r$ ). This would give social welfare of $\ln(n)$ , whereas the optimal assignment would give $n(1-\epsilon)$ . Without information about the game state, therefore, the players may achieve only a $O\left(\frac{\ln(n)}{n}\right)$ fraction of the possible welfare.

1.2 Information Model

In resource sharing games, players’ decisions about their actions will be best when they know how many players have chosen each resource when they arrive. The mechanisms we consider, therefore, will publicly announce some estimate of these counts. We consider the trade-off between the privacy lost by publishing these estimates and the accuracy of the counters in terms of social welfare. We consider three categories of counters for publicly posting the estimate of resource usage: perfect, private and empty counters.

Perfect Counters: At all points, the counters display the exact usage of each resource.

Privacy-preserving public counters: At all points, the counters display an approximate usage of the resources while maintaining privacy for each player. We define the privacy guarantee in Section 2.

Empty Counters: At all points, every counter displays the value 0.

1.3 Players’ Behavior

Each player is a utility-maximizing agent and will choose the resource that, given their beliefs about actions taken by previous players and the publicly displayed counters, gives them maximum value. We analyze the game play under two classes of strategies – greedy and undominated strategies.

1.

Greedy strategy: Under the greedy strategy, a player has no outside belief about the actions of previous players and chooses the resource that maximizes her utility given the currently displayed (or announced) values of the counters. Greedy is a natural choice of strategy to consider since it is the utility-maximizing strategy when the usage counts posted are perfect.
2.

Undominated Strategy(UD): Under undominated strategies, we allow players to have any beliefs about the actions of the previous players that are consistent with the displayed value of the counters⁴⁴4As will become clear in Section 2, we work with privacy-preserving public counters that display values that can be off from the true usage only in a bounded range. Hence with these counters, a player’s belief is consistent as long as the belief implies the usage of the resource to be a number that is within the bounded range of the displayed value. Moreover, with empty counters, any belief about the actions of previous players is a consistent belief., and they are allowed to play any undominated strategy $a_{i}$ under this belief. A strategy $a_{i}$ is undominated under a belief, if no other $a^{\prime}_{i}$ get a strictly higher utility. ⁵⁵5For each counter mechanism we consider, there exists at least one undominated strategy. For example, with perfect counters, the only consistent belief is that the true value is equal to the displayed value and here the greedy strategy is always undominated; moreover, if the counter mechanism has a nonzero probability of outputting the true value, then again the greedy strategy is undominated under the belief that the displayed value is the true value; if the counter mechanism can display values that are arbitrarily off from the true value, then for equal initial values every strategy is undominated.

We analyze the social welfare $SW(a)=\sum_{i}u_{i}(a)$ generated by an announcement mechanism $\mathcal{M}$ for a set of strategies $D$ and compare it to the optimal social welfare $OPT$ . For a game setting $g$ , constituted of a collection of players $[n]$ and their allowable actions $A_{i}$ (as defined in Section 1.1), $OPT(g)$ is defined as the optimal social welfare that can be achieved by any allocation of resources to the players, where the space of feasible allocations is determined by the setting $g$ . In the unit-demand setting, $OPT(g)$ is the maximum weight matching in the bipartite graph $G=(U\cup V,E)$ where $U$ is the set of the $n$ players, $V$ has $n$ vertices for each resource $r$ , one of value $V_{r}(k)$ for each $k\in[n]$ , and there is an edge between player $i$ and all vertices corresponding to resource $r$ if and only if $r\in A_{i}$ (Note that the weights are on the vertices in $V$ ). The object of our study is $CR_{D}(g,\mathcal{M})$ , the worst case competitive ratio of the optimal social welfare to the welfare achieved under strategy $D$ and counter mechanism $\mathcal{M}$ . As mentioned earlier, $D$ will either be the greedy (Greedy) or the undominated (Undom) strategy, and $\mathcal{M}$ will be either the perfect ( $\mathcal{M}_{Full}$ ), the privacy-preserving or the empty ( $\mathcal{M}_{\emptyset}$ ) counter. When $\mathcal{M}$ uses internal random coins, our results will either be worst-case over all possible throws of the random coins, or will indicate the probability with which the social welfare guarantee holds.

1.4 Statement of Main Results

For sequential resource-sharing games, we prove that for all nonincreasing value curves, the greedy strategy following privacy-preserving counters has a competitive ratio polylogarithmic in the number of players (Theorem 5). This should be contrasted with the competitive ratio of $4$ achieved by greedy w.r.t. perfect counters (Theorem 1) and the linear (in the number of players) competitive ratio of greedy with empty counters (as shown in the illustrative example in Section 1.1). For the case of undominated strategies, when the marginal values of resources drop slowly, (for example, at a polynomial rate, $V_{r}(k)=V_{r}(0)/k^{p}$ for constant $p>0$ ), we bound the competitive ratio (w.r.t. privacy-preserving counters) (Theorem 7). With empty counters, the competitive ratio for undominated strategies is unbounded (Theorem 2) for arbitrary curves and is at least quadratic (in the number of players) if the value curve drops slowly (Theorem 3). We note here that for many of our positive results for privacy preserving counters state the competitive ratio in terms of parameters of the counter vector $\alpha$ and $\beta$ (as detailed in Section 2) and for a particular implementation of the counter vectors, the values of $\alpha$ and $\beta$ are mentioned in Section 4.

The key privacy tool we use is the differentially private counter under continual observation [4], which we use to publish estimates of the usage of each resource. We improve upon the existing error guarantees of differentially private counters and design a new differentially private counter in Section 4. The new counter provides a tighter additive guarantee at the price of introducing a constant multiplicative error.

In Section 5, we consider other classes of games – specifically, we analyze Unrelated Machine Scheduling, Cut, and Cost Sharing games. The work of Leme et al. [12] showed these games have improved sequential price of anarchy over the simultaneous price of anarchy. For these games, we ask the question: if players do not have perfect information to make decisions, but instead have only noisy approximations (due to privacy considerations), does sequentiality still improve the quality of play? We prove that the answer is affirmative in most cases, and furthermore, for some instances, having differentially-private information dissemination improves the competitive ratio over perfect information (Proposition 3).

1.5 Related Work

A great deal of work has been done at the intersection of mechanism design and privacy; Pai and Roth [15] have an extensive survey. Our work is similar to much of the previous work in that it considers maintaining differential privacy to be a constraint. The focus of our work however is on how useful information can be provided to players in games of imperfect information to help achieve a good social objective while respecting the privacy constraint of the players. The work of Kearns et al. [11] is close in spirit to ours. Kearns et al. [11] consider games where players have incomplete information about other players’ types and behaviors. They construct a privacy-preserving mechanism which collects information from players, computes an approximate correlated equilibria, and then advises players to play according to this equilibrium. The mechanism is approximately incentive compatible for the players to participate in the mechanism and to follow its suggestions. Several later papers [16, 9] privately compute approximate equillibria in different settings. Our main privacy primitive is the differentially private counters under continual observation [4, 3], also used in much of the related work on private equilibrium computation.

Our investigation of cut games, unrelated machine scheduling, and cost-sharing (Section 5) is inspired by work of Leme et al. [12]. Their work focuses on the improvement in social welfare of equilibria in the sequential versus the simultaneous versions of certain games. We ask a related question: when we consider sequential versions of games, and only private, approximate information about the state of play (as opposed to perfect) is given to players, how much worse can social welfare be?

As mentioned in Section 1.3, one class of player behavior for which we analyze the games is greedy. Our analysis of greedy behavior is in part inspired by the work of Balcan et al. [2], who study best response dynamics with respect to noisy cost functions for potential games. An important distinction between their setting and ours is that the noisy estimates we consider are estimates of state, not value, and may for natural value curves be quite far from correct in terms of the values of the actions.

2 Privacy-preserving public counters

We design announcement mechanisms $\mathcal{M}_{i}$ which give approximate information about actions made by the previous players to player $i$ . Let $\Delta_{m}$ denote the action space for each player (the $m$ -dimensional simplex $\Delta_{m}=\{a\in[0,1]^{m}\mid\|a\|_{1}\leq 1\}$ ). Mechanism $\mathcal{M}_{i}:\left({\Delta_{m}}\right)^{i-1}\times R\to{\Delta_{m}}$ depends upon the actions taken before $i$ (specifically, the usage of each resource by each player), and on internal random coins $R$ . When player $i$ arrives, $m_{i}(a_{1},\ldots,a_{i-1})\sim\mathcal{M}_{i}(a_{1},\ldots,a_{i-1})$ is publicly announced. Player $i$ plays according to some strategy $d_{i}:{\Delta_{m}}\to A_{i}$ , that is $a_{i}=d_{i}(m_{1},\ldots,m_{i}(a_{1},\ldots,a_{i-1}))$ , a random variable which is a function of this announcement. When it is clear from context, we denote $m_{i}(a_{1},\ldots,a_{i-1})$ by $m_{i}$ . Formally, the counters used in this paper satisfy the following notion of privacy.

Definition 1.

An announcement mechanism $\mathcal{M}$ is $(\epsilon,\delta)$ -differentially private under adaptive⁶⁶6Adaptivity is needed in this case because the announcements are arguments to the actions of players: when a particular action changes, this modifies the distribution over the future announcements, which in turn changes the distribution over future selected actions. continual observation in the strategies of players if, for each $d$ , for each player $i$ , each pair of strategies $d_{i},d^{\prime}_{i}$ , and every $S\subseteq({\Delta_{m}})^{n}$ :

\mathbb{P}[(m_{1},\ldots,m_{n})\in S]\leq e^{\epsilon}\mathbb{P}[(m_{1},\ldots,m_{i},m_{i+1}^{\prime}\ldots,m_{n}^{\prime})\in S]+\delta

where $m_{j}\sim\mathcal{M}_{j}(a_{1},\ldots,a_{j-1})$ and $m_{j}^{\prime}\sim\mathcal{M}_{j}(a_{1},\ldots,a_{i-1},a^{\prime}_{i},a^{\prime}_{i+1},\ldots,a^{\prime}_{j-1})$ , $a_{j}=d_{j}(m_{1},\ldots,m_{j})$ , and $a^{\prime}_{i}=d^{\prime}_{i}(m_{1},\ldots,m_{i})$ , and for all $j>i$ , $a^{\prime}_{j}=d_{j}(m_{1},\ldots,m_{i-1},m_{i},m^{\prime}_{i+1},\ldots,m^{\prime}_{j})$ .

This definition requires that two worlds which differ in a single player changing her strategy from $d_{i}$ to $d^{\prime}_{i}$ have statistically close joint distributions over all players’ announcements (and thus their joint distributions over actions). Note that the distribution of $j>i$ ’s announcement can change slightly, causing $j$ ’s distribution over actions to change slightly, necessitating the cascaded $m^{\prime}_{j},a^{\prime}_{j}$ for $j>i$ in our definition. The mechanisms we use maintain approximate use counters for each resource. The values of the counters are publicly announced throughout the game play. We now define the notion of accuracy used to describe these counters.

Definition 2 ( $(\alpha,\beta,\gamma)$ -accurate counter vector).

A set of counters $y_{i,r}$ is defined to be $(\alpha,\beta,\gamma)$ -accurate if with probability at least $1-\gamma$ , at all points of time, the displayed value of every counter $y_{i,r}$ lies in the range $[\frac{x_{i,r}}{\alpha}-\beta,\alpha x_{i,r}+\beta]$ where $x_{i,r}$ is the true count for resource $i$ , and is monotonically increasing in the true count.

We refer to a set of $(\alpha,\beta,0)$ -accurate counters as $(\alpha,\beta)$ -counters for brevity. It is possible to achieve $\gamma=0$ (which is necessary for undominated strategies, which assumes the multiplicative and additive bounds on $y$ are worst-case), taking an appropriate loss in the privacy guarantees for the counter (Proposition 1). Counters satisfying Definitions 1 and 2 with $\alpha=1$ and $\beta=O(\log^{2}n)$ were given in Dwork et al. [4], Chan et al. [3]; we give a different implementation in Section 4 which gives a tighter bound on $\alpha\beta$ by taking $\alpha$ to be a small constant larger than 1. Furthermore, the counters in Section 4 are monotonic (i.e., the displayed values can only increase as the game proceeds) and we use monotonicity of the counters in some of our results.

In some settings we require counters we a more specific utility guarantee:

Definition 3 ( $(\alpha,\beta,\gamma)$ -accurate underestimator counter vector).

A set of counters $y_{i,r}$ is defined to be $(\alpha,\beta,\gamma)$ -accurate if with probability at least $1-\gamma$ , at all points of time, the displayed value of every counter $y_{i,r}$ lies in the range $[\frac{x_{i,r}}{\alpha}-\beta,x_{i,r}]$ where $x_{i,r}$ is the true count for resource $i$ .

The following observation states that a counter vector can be converted to an undercounter with small loss in accuracy.

Observation 1.

We can convert a $(\alpha,\beta)$ -counter to an $\left(\alpha^{2},\frac{2\beta}{\alpha}\right)$ -underestimating counter vector.

Proof.

We can shift the counter, $\frac{1}{\alpha}x-\beta\leq y\leq\alpha x+\beta$ implies $y^{\prime}=\frac{y-\beta}{\alpha}\leq x$ and $\frac{1}{\alpha^{2}}x-\frac{2\beta}{\alpha}\leq y^{\prime}$ . ∎

3 Resource Sharing

In this section, we consider resource sharing games – the utility to a player is completely determined by the resource she chooses and the number of players who have chosen that resource before her. This section considers the case where players’ actions are discrete: $a_{i}\in\{0,1\}^{m}$ for all $i,a_{i}\in A_{i}$ . We defer the analysis of the case where players’ actions are continuous to Appendix C.

3.1 Perfect counters and empty counters

Before delving into our main results, we point out that, with perfect counters, greedy is the only undominated strategy, and the competitive ratio of greedy is a constant. We state this result formally, and defer its proof to Appendix B.

Theorem 1.

With perfect counters, greedy behavior is dominant-strategy and all other behavior is dominated for any sequential resource-sharing game $g$ ; furthermore, $CR_{\textsc{Greedy}}(\mathcal{M}_{Full},g)=4$ .

Recall, from our example in the introduction, that both greedy and undominated strategies can perform poorly with respect to empty counters. We defer the proof of the following results to Appendix B. Recall that $\mathcal{M}_{\emptyset}$ refers to the empty counter mechanism.

Theorem 2.

There exist games $g$ such that $CR_{\textsc{Undom}}(\mathcal{M}_{\emptyset},g)$ is unbounded.

Theorem 3.

There exists $g$ such that $CR_{\textsc{Undom}}(\mathcal{M}_{\emptyset},g)\geq\Omega(\frac{n^{2}}{\log(n)})$ , when $V_{r}(t)=\frac{V_{r}(0)}{t}$ .

3.2 Privacy-preserving public counters and Greedy Strategy

Theorem 4.

With $(\alpha,\beta)$ -accurate underestimator counter mechanism $\mathcal{M}$ , $CR_{\textsc{Greedy}}(\mathcal{M},g)=O(\alpha\beta)$ for all resource-sharing game $g$ .

Before we prove Theorem 4, we need a way to compare players’ utilities with the utility they think they get from choosing resources greedily with respect to approximate counters. Let a player’s perceived value be $V_{r}(y_{i,r})$ where $r$ is the resource she chose (the value of a resource if the counter was correct, which may or may not be the actual value of the resource).

Lemma 1.

Suppose players choose greedily according to a $(\alpha,\beta)$ -underestimator. Then, the sum of their actual values is at least a $\frac{1}{2\alpha\beta}$ -fraction of the sum of their perceived values.

Proof.

Suppose $k$ players chose a given resource $r$ . For ease of notation, let these be players $1$ through $k$ . We wish to bound the ratio

\frac{\sum_{i=1}^{k}V_{r}(y_{i,r})}{\sum_{c=1}^{k}V_{r}(c)}.

We start by “grouping” the counter values: it cannot take on values that are small for more than a certain number of steps. In particular, if $x_{i,r}>T\alpha\beta$ , for some $T\in\mathbb{N}$ ,

\displaystyle y_{i,r}\geq

\displaystyle\frac{1}{\alpha}x_{i,r}-\beta\geq\frac{T\alpha\beta}{\alpha}-\beta=\left(T-1\right)\beta

Now, we bound the ratio from above using this fact.

\displaystyle\frac{\sum_{i=1}^{k}V_{r}(y_{i,r})}{\sum_{c=1}^{k}V_{r}(c)}

\displaystyle\leq\frac{2\alpha\beta\sum_{T=1}^{\lceil\frac{k}{\alpha\beta}\rceil}V_{r}((T-1)\beta)}{\sum_{c=1}^{k}V_{r}(c)}\leq\frac{2\alpha\beta\sum_{T=1}^{\lceil\frac{k}{\alpha\beta}\rceil}V_{r}((T-1)\beta)}{\sum_{T=1}^{\lceil\frac{k}{\alpha\beta}\rceil}V_{r}((T-1)\beta)}\leq 2\alpha\beta

where the first inequality came from the fact that the value curves are non-increasing and the lower bound on the counter values from above, and the second because all terms are nonnegative. ∎

Proof of Theorem 4.

The optimal value of the resource-sharing game $g$ , denoted by $OPT(g)$ , is the maximum weight matching in the bipartite graph $G=(U\cup V,E)$ where $U$ is the set of the $n$ players and $V$ has $n$ vertices for each resource $r$ , one of value $V_{r}(k)$ for each $k\in[n]$ . There is an edge between player $i$ and all vertices corresponding to resource $r$ if and only if $r\in A_{i}$ . Note that the weights are on the vertices in $V$ .

We now define a complete bipartite graph $G^{\prime}$ which has the same set of nodes but whose node weights differ for some nodes in $G$ . Consider some resource $r$ , and the collection of players who chose $r$ in $g$ . If there were $t_{k}$ players $i$ who chose resource $r$ when $y_{i,r}=k$ , make $t_{k}$ of the nodes corresponding to $r$ have weight $V_{r}(k)$ . Finally, if there were $F_{k}$ players who chose resource $r$ , let the remaining $n-F_{k}$ nodes corresponding to $r$ have weight $V_{r}(F_{k}+1)$ .

We first claim that the perceived utility of players choosing greedily according to the counters is identical to the weight of the greedy matching in $G^{\prime}$ (where nodes arrive in the same order). We prove, in fact, that the corresponding matching will be identical by induction. Since the counters are monotone, earlier copies of a resource appear more valuable. So, when the first player arrives in $G^{\prime}$ , the most valuable node she has access to is exactly the first node corresponding to the resource she took according to the counters. Now, assume that prior to player $i$ , all players have chosen nodes corresponding to the resource they chose according to the counters. By our induction hypothesis and monotonicity of the counters and value curves, there is a node $n_{i}$ corresponding to $i$ ’s selection $r$ according to counters of weight $V_{r}(y_{i,r})$ , and no heavier node corresponding to $r$ . Likewise, for all other resources $r^{\prime}$ , all nodes corresponding to $r^{\prime}$ have weight more than $V_{r^{\prime}}(y_{i,r^{\prime}})$ . Thus, $i$ will take $n_{i}$ for value $V_{r}(y_{i,r})$ . Thus, the weight of the greedy matching in $G^{\prime}$ equals the perceived utility of greedy play according to the counters.

Let $\textsc{Greedy}_{\textsc{counters}}$ denote the set of actions players make playing greedily with respect to the counters. By Lemma 1, the social welfare of $\textsc{Greedy}_{\textsc{counters}}$ is a $\frac{1}{\alpha\beta}$ -fraction of the perceived social welfare. By our previous argument, the perceived social welfare of greedy play according to the counters is the same as the weight of the greedy matching in $G^{\prime}$ . By Theorem 1, the greedy matching in $G^{\prime}$ is a $4$ -approximation to the max-weight matching in $G^{\prime}$ . Finally, since the counters are underestimators, the weight of the max-weight matching in $G^{\prime}$ is at least as large as $OPT(g)$ . Thus, we know that the social welfare of greedy play with respect to counters is a $\frac{1}{2\alpha\beta}$ fraction of the optimal social welfare to $g$ . ∎

Theorem 5.

There exists $(\epsilon,\delta)$ -privacy-preserving mechanism $\mathcal{M}$ such that

CR_{\textsc{Greedy}}(\mathcal{M},g)=\min\left(O\left(\frac{(\log n)(\log(nm/\delta))}{\epsilon}\right),O\left(\frac{m\log n\log\log(1/\delta)}{\epsilon}\right)\right)

for all resource-sharing games $g$ .

Proof.

In Section 4, we prove Corollary 2 that says that we can achieve an $(\epsilon,\delta)$ -differentially private counter vector achieving the better of $(1,O(\frac{(\log n)(\log(nm/\delta))}{\epsilon}))$ -accuracy and $(\alpha,\tilde{O}_{\alpha}(\frac{m\log n\log\log(1/\delta)}{\epsilon}))$ -accuracy for any constant $\alpha>1$ . This along with Theorem 4 proves the result. ∎

In Appendix B.1, Observation 3 proves that players acting greedily according to any estimate that is deterministically more accurate than the values provided by the private counters also achieve similar or better social welfare guarantees. Moreover, we show that if the estimates used by the players are more accurate only in expectation, as opposed to deterministically, then we cannot make a similar claim (Observation 4).

3.3 Privacy-preserving public counters and Undominated strategies

We begin with an illustration of how undominated strategies can perform poorly for arbitrary value curves, as motivation for the restricted class of value curves we consider in Theorem 7. In the case of greedy players, we were able to avoid the problem of players undervaluing resources rather easily, by forcing the counters to only underestimate $x_{i,r}$ . This won’t work for undominated strategies: players who know the counts are shaded downward can compensate for that fact.

Theorem 6.

For an $(\epsilon,\delta)$ –differentially private announcement mechanism $\mathcal{M}$ , there exist games $g$ for which $CR_{\textsc{Undom}}(g,\mathcal{M})=\Omega\left(\frac{1}{\delta}\right)$ .

Proof.

Suppose there are two players $1$ and $2$ , and resources $r,r^{\prime}$ . Let $r$ have $V_{r}(0)=1$ , $V_{r}(1)=0$ , and $V_{r^{\prime}}(k)=\rho$ , for all $k\geq 0$ . Furthermore, let player $1$ have access only to resource $r^{\prime}$ but player $2$ has access to both $r$ and $r^{\prime}$ . Player $1$ will choose $r^{\prime}$ . Let player $2$ ’s strategy be $d_{2}$ , such that if she determines there was nonzero chance that player $1$ chose $r$ according to her signal $m_{2}$ , she will choose resource $r^{\prime}$ . This is undominated: if $1$ did choose $r$ , $r^{\prime}$ will be more valuable for $2$ . Thus, if $2$ sees any signal that can occur when $r$ is chosen by $1$ , she will choose $r^{\prime}$ . The collection of signals $2$ can see if $1$ chooses $r$ has probability $1$ in total. So, because $m_{2}$ is $(\epsilon,\delta)$ -differentially private in player $1$ ’s action, the set of signals reserved for the case when $1$ chooses $r^{\prime}$ (that cannot occur when $r$ is chosen by $1$ ) may occur with probability at most $\delta$ (they can occur with probability $0$ if $1$ chose $r$ , implying they can occur with probability at most $\delta$ when $1$ chooses $r^{\prime}$ ). Thus, with this probability $1-\delta$ , player $2$ will choose $r^{\prime}$ , implying $\mathbb{E}[SW]\leq(1-\delta)2\rho+\delta(1+\rho)=\delta+(2-\delta)\rho$ , which for $\rho$ sufficiently small approaches $\delta$ , while $1+\rho$ is the optimal social welfare. ∎

Given the above example, we cannot hope to have a theorem as general as Theorem 4 when analyzing undominated strategies with privacy-preserving counters. Instead, we show that, for a class of well-behaved value curves, we can bound the competitive ratio of undominated strategies (Theorem 7).

Again, along the lines of the greedy case, we show that any player who chooses any undominated resource $r^{\prime}$ over resource $r$ gets a reasonable fraction of the utility she would get from choosing $r$ . Then, by the analysis of greedy players, we have an analogous argument implying the bound of Theorem 7.

Theorem 7.

If each value curve $V_{r}$ has the property that $\psi(\alpha,\beta)V_{r}(x)\geq V_{r}((\max\{0,\frac{x}{\alpha^{2}}-\frac{2\beta}{\alpha}\}))$ and also $V_{r}((\alpha^{2}x+2\alpha\beta))\geq\phi(\alpha,\beta)V_{r}(x)$ , then an action profile $a$ of undominated strategies according to $(\alpha,\beta)$ -counter vector $\mathcal{M}$ has $CR_{\textsc{Undom}}(g,\mathcal{M})=O\left(\psi(\alpha,\beta)\phi(\alpha,\beta)\right)$ .

In particular, Theorem 7 shows that, for games where $V_{r}(i)=\frac{V_{r}(0)}{g_{r}(x_{i,r})}$ , where $g_{r}$ is a polynomial, the competitive ratio of undominated strategies degrades gracefully as a function of the maximum degree of those polynomials. A simple calculation implies the following corollary, whose proof we relegate to Appendix B.3.

Corollary 1.

Suppose for a resource-sharing game $g$ , each resource $r$ has a value curve of the form $V_{r}(x)=\frac{V_{r}(0)}{g_{r}(x)}$ , where $g_{r}$ is a monotonically increasing degree- $d$ polynomial and $V_{r}(0)$ is some constant. Then, $CR_{\textsc{Undom}}(g,\mathcal{M})\leq O(2\alpha^{3}\beta)^{d}$ with $\mathcal{M}$ providing $(\alpha,\beta)-$ counters.

4 Private Counter Vectors with Lower Errors for Small Values

In this section, we describe a counter for the model of differential privacy under continual observation that has improved guarantees when the value of the counter is small. Recall the basic counter problem: given a stream ${\vec{a}}=(a_{1},a_{2},...,a_{n})$ of numbers $a_{i}\in[0,1]$ , we wish to release at every time step $t$ the partial sum $x_{t}=\sum_{i=1}^{t}a_{i}$ . We require a generalization, where one maintains a vector of $m$ counters. Each player’s update contribution is now a vector $a_{i}\in\Delta_{m}=\{a\in[0,1]^{m}\mid\|a\|_{1}\leq 1\}$ . That is, a player can add non-negative values to all counters, but the total value of her updates is at most 1. The partial sums $x_{t}$ then lie in $(\mathbb{R}^{+})^{m}$ and have $\ell_{1}$ norm at most $t$ .

Given an algorithm $\mathcal{M}$ , we define the output stream $(y_{1},...,y_{n})=\mathcal{M}({\vec{a}})$ where $y_{t}=\mathcal{M}(t,a_{1},...,a_{t-1})$ lies in $\mathbb{R}^{m}$ . We seek counters that are private (Definition 1) and satisfy a mixed multiplicative and additive accuracy guarantee (Definition 2). Proofs of all the results in this section can be found in Appendix E.

The original works on differentially private counters [4, 3] concentrated on minimizing the additive error of the estimated sums, that is, they sought to minimize $\|x_{t}-y_{t}\|_{\infty}$ . Both papers gave a binary tree-based mechanism, which we dub “TreeSum”, with additive error approximately $(\log^{2}n)/\epsilon$ . Some of our algorithms use TreeSum, and others use a new mechanism (FTSum, described below) which gets a better additive error guarantee at the price of introducing a small multiplicative error. Formally, they prove:

Lemma 2.

For every $m\in\mathbb{N}$ and $\gamma\in(0,1)$ : Running $m$ independent copies of TreeSum [4, 3] is $(\epsilon,0)$ -differentially private and provides an $(1,C_{tree}\cdot\frac{(\log n)(\log(nm/\gamma))}{\epsilon},\gamma)$ -approximation to partial vector sums, where $C_{tree}>0$ is an absolute constant.

Even for $m=1,\alpha=1$ , this bound is slightly tighter than those in Chan et al. [3] and Dwork et al. [4]; however, it follows directly from the tail bound in Chan et al. [3].

Our new algorithm, FTSum (for Flag/Tree Sum), is described in Algorithm 1. For small $m$ ( $m=o(log(n))$ ), it provides lower additive error at the expense of introducing an arbitrarily small constant multiplicative error.

Lemma 3.

For every $m\in\mathbb{N}$ , $\alpha>1$ and $\gamma\in(0,1)$ , FTSum (Algorithm 1) is $(\epsilon,0)$ -differentially private and $(\alpha,\tilde{O}_{\alpha}(\frac{m\log(n/\gamma)}{\epsilon}),\gamma)$ -approximates partial sums (where $\tilde{O}_{a}(\cdot)$ hides polylogarithmic factors in its argument, and treats $\alpha$ as constant).

FTSum proceeds in two phases. In the first phase, it increments the reported output value only when the underlying counter value has increased significantly. Specifically, the mechanism outputs a public signal, which we will call a “flag”, roughly when the true counter achieves the values $\log n$ , $\alpha\log n$ , $\alpha^{2}\log n$ and so on, where $\alpha$ is the desired multiplicative approximation. The reported estimate is updated each time a flag is raised (it starts at 0, and then increases to $\log n$ , $\alpha\log n$ , etc). The privacy analysis for this phase is based on the “sparse vector” technique of Hardt and Rothblum [8], which shows that the cost to privacy is proportional to the number of times a flag is raised (but not the number of time steps between flags).

When the value of the counter becomes large (about $\frac{\alpha\log^{2}n}{(\alpha-1)\epsilon}$ ), the algorithm switches to the second phase and simply uses the TreeSum protocol, whose additive error (about $\frac{\log^{2}n}{\epsilon}$ ) is low enough to provide an $\alpha$ multiplicative guarantee (without need for the extra space given by the additive approximation).

If the mechanism were to raise a flag exactly when the true counter achieved the values $\log n$ , $\alpha\log n$ , $\alpha^{2}\log n$ , etc, then the mechanism would provide a $(\alpha,\log n,0)$ approximation during the first phase, and a $(\alpha,0,0)$ approximation thereafter. The rigorous analysis is more complicated, since flags are raised only near those thresholds.

Input: Stream

{\vec{a}}=(a_{1},...,a_{n})\in([0,1]^{m})^{n}

, parameters

m,n\in\mathbb{N}

\alpha>1

and

\gamma>0

Output: Noisy partial sums

y_{1},...,y_{n}\in\mathbb{R}^{m}

k\leftarrow\lceil\log_{\alpha}(\frac{\alpha}{\alpha-1}\cdot C_{tree}\cdot\frac{\log(nm/\gamma)}{\epsilon})\rceil

;

C_{tree}

is the constant from Lemma 2 */

\epsilon^{\prime}\leftarrow\frac{\epsilon}{2m(k+1)}

;

for $r=1$ to $m$ do

\text{flag}_{r}\leftarrow 0

;

x_{0,r}\leftarrow 0

;

\tau_{r}\leftarrow(\log n)+{\mathsf{Lap}}(2/\epsilon^{\prime})

;

for $i=1$ to $n$ do

for $r=1$ to $m$ do

if $\text{flag}_{r}\leq k$ then ( First phase still in progress for counter

r

)

x_{i,r}\leftarrow x_{i-1,r}+a_{i,r}

;

\tilde{x_{i,r}}\leftarrow x_{i,r}+{\mathsf{Lap}}(\frac{2}{\epsilon^{\prime}})

;

if $\tilde{x_{i,r}}>\tau_{r}$ then ( Raise a new flag for counter

r

)

\text{flag}_{r}\leftarrow\text{flag}_{r}+1

;

\tau_{r}\leftarrow(\log n)\cdot\alpha^{\text{flag}_{r}}+{\mathsf{Lap}}(2/\epsilon^{\prime})

;

Release

y_{i,r}=(\log n)\cdot\alpha^{\text{flag}_{r}-1}

;

else ( Second phase has been reached for counter

r

)

Release

y_{i,r}=

r

-th counter output from TreeSum

({\vec{a}},\epsilon/2)

);

Algorithm 1 FTSum — A Private Counter with Low Multiplicative Error

Enforcing Additional Guarantees

Finally, we note that it is possible to enforce to additional useful properties of the counter. First, we may insist that the accuracy guarantees be satisfied with probability 1 (that is, set $\gamma=0$ ), at the price of increasing the additive term $\delta$ in the privacy guarantee:

Proposition 1.

If $\mathcal{M}$ is $(\epsilon,\delta)$ -private and $(\alpha,\beta,\gamma)$ -accurate, then one can modify $\mathcal{M}$ to obtain an algorithm $\mathcal{M}^{\prime}$ with the same efficiency that is $(\epsilon,\delta+\gamma)$ -private and $(\alpha,\beta,0)$ -accurate.

Second, as in [4], we may enforce the requirement that the reported values be monotone, integral values that increase at each time step by at most 1. The idea is to simply report the nearest integral, monotone sequence to the noisy values (starting at 0 and incrementing the reported counter only when the noisy value exceeds the current counter).

Proposition 2 ([4]).

If $\mathcal{M}$ is $(\epsilon,\delta)$ -private and $(\alpha,\beta,\gamma)$ -accurate, then one can modify $\mathcal{M}$ to obtain an algorithm $\mathcal{M}^{\prime}$ which reports monotone, integral values that increase by 0 or 1 at each time step, with the same privacy and accuracy guarantees as $\mathcal{M}$ .

Corollary 2.

Algorithm 1 is an $(\epsilon,\delta)$ -differentially private vector counter algorithm providing a

1.

$(1,O(\frac{(\log n)(\log(nm/\delta))}{\epsilon}),0)$ -approximation (using modified TreeSum); or
2.

$(\alpha,\tilde{O}_{\alpha}(\frac{m\log n\log\log(1/\delta)}{\epsilon}),0)$ -approximation for any constant $\alpha>1$ (using FTSum).

5 Extensions

As part of Appendix D.2, we also consider settings where players’ utility when choosing a resource $r$ depends upon the total number of players choosing $r$ , not just the players who chose $r$ before. In addition, Appendix A considers several other classes of games: namely, cut games, consensus games, and unrelated machine scheduling, and consider whether or not private synopses of the state of play is sufficient to improve social welfare over simultaneous play, as perfect synopses have been proven to be in Leme et al. [12].

6 Discussion and Open Problems

In this work, we considered how public dissemination of information in sequential games can guarantee a good social welfare while maintaining differential privacy of the players’ strategies. We considered two ‘extreme’ cases – the greedy strategy and the class of all undominated strategies. While analyzing the class of undominated strategies gives guarantees that are robust, in many games that we considered, the competitive ratios were significantly worse than greedy strategies, and in some cases they were unbounded. It is interesting to note that many of the examples in this paper that show the poor performance with undominated strategies also hold when the players know their position in the sequence, an assumption we have not made for any of the positive results in this work. It is an interesting direction for future research to consider classes of strategies that more restricted than undominated strategies yet are general enough to be relevant for games where players play with imperfect information.

As mentioned in the introduction, we note here that, while players are making choices subject to approximate information, our results are not a direct extension of the line of thought that approximate information implies approximate optimization. In particular, for greedy strategies, while there may be a bound on the error of the counters, but that does not imply that, for arbitrary value curves, playing greedily according to the counters will be approximately optimal for each individual. In particular, consider one resource $r$ with value $H$ for the first $10$ investors, and value $0$ for the remaining investors, and a second resource $r^{\prime}$ with value $H/2$ for all investors. With $(\alpha,\beta,\gamma)$ , as many as $\beta$ players might have unbounded ratio between their value for $r$ as $r^{\prime}$ , but will pick $r$ over $r^{\prime}$ . The analysis of greedy shows, despite this anomaly, the total social welfare is still well-approximated by this behavior.

All of our results relied on using differentially private counters for disseminating information. For the differentially-private counter, a main open question is “what is the optimal trade-off between additive and multiplicative guarantees?”. Furthermore, as part of future research, one can consider other privacy techniques for announcing information that can prove useful in helping players achieve a good social welfare. And more generally, we want to understand what features of games lend themselves to be amenable to public dissemination of information that helps achieve good welfare and simultaneously preserves privacy of the players’ strategies.

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Appendix A Other games

In this section, we study a number of games which Leme et al. [12] showed to have a large improvement between their Price of Anarchy and their sequential Price of Anarchy. We pose the question: with privacy-preserving information handed out to players, what loss is incurred in comparison to providing exact information? In addition to introducing privacy constraints, we should note here that while in Leme et al. [12], each player playing the sequential game knows the type of every other player, in our setting, we only provide information about actions taken by previous players.

A.1 Unrelated Machine scheduling games

An instance of the unrelated machine scheduling game consists of $n$ players who must schedule their respective jobs on one of the $m$ machines; the cost to the player is the final load on the machine on which she scheduled her job. The size of player $k$ ’s job on machine $q$ is $t_{kq}$ . The objective of the mechanism is to minimize the makespan. We consider the dynamics of this game when it is played sequentially. Leme et al. [12] prove that the sequential price of anarchy is $O(m2^{n})$ .

In our setting, each player is shown a load profile when it is her turn to play. The load profile $L$ denotes the displayed vector of loads on the various machines. In the perfect counter setting, the displayed load equals the exact load on each machine. We now show that undominated strategies, with perfect counters, perform unboundedly poorly with respect to OPT.

Lemma 4.

If $\mathcal{M}$ is a perfect counter vector, $CR_{\textsc{Undom}}(\mathcal{M},g)$ is unbounded for some instances $g$ of unrelated machine scheduling.

Proof.

Consider the case with two players ( $p_{1}$ and $p_{2}$ ) and two machines ( $m_{1}$ and $m_{2}$ ). $p_{1}$ arrives before $p_{2}$ . Player $p_{1}$ has a cost of 0 on $m_{1}$ and 1 on $m_{2}$ . It is an undominated strategy for player 1 to choose $m_{2}$ since if player $p_{2}$ has a cost of 2 on $m_{1}$ and 3 on $m_{2}$ , $p_{2}$ chooses $m_{1}$ and so $p_{1}$ is better off scheduling her job on $m_{2}$ .

However, if $p_{1}$ chooses $m_{2}$ (an undominated strategy) $m_{2}$ , and player $p_{2}$ has cost 1 on $m_{1}$ and 0 on $m_{2}$ , the optimal makespan is 0; the achieved makespan is at least 1. ∎

In light of this result, we restrict our attention to greedy strategies for machine scheduling, and show that the competitive ratio of the greedy strategy with privacy-preserving counters is bounded. Below, we denote by $t^{*}_{k}$ the minimum cost of job $k$ among all the machines and by $q^{*}_{k}$ the machine that achieves this minimum. The following result follows from the analysis of the greedy algorithm as presented in Aspnes et al. [1].

Theorem 8.

[1] With perfect counters and players playing greedy strategies, the makespan is at most $\sum_{i=1}^{n}t_{i}^{*}$ , and since $OPT\geq\sum_{i=1}^{n}t_{i}^{*}/m$ , the competitive ratio is at most $m$ .

Theorem 9 shows that such a bound extends to the setting where players have only approximate information about the state, showing that privacy-preserving information is enough to attain nontrivial coordination with greedy players.

Theorem 9.

Using $(\alpha,\beta,\gamma)$ -counter vector, and players playing greedy strategies, with probability $1-\gamma$ , the makespan is at most $\alpha^{2n+1}m\cdot OPT+\beta(\alpha^{2n+1}(2n+1)+1)$ .

Proof of Theorem 9.

Consider any player $i$ , and let the displayed load profile she sees be $L$ . Using greedy strategy, she will put her job on machine $q$ that minimizes $L_{q}+t_{kq}$ and this in particular shall be at most $\lvert L\rvert_{\infty}+t^{*}_{k}$ . Since the true makespan before this player placed her job is at most $\alpha\lvert L\rvert_{\infty}+\beta$ , hence after she places her job, for the displayed load profile $L^{\prime}$ , $\lvert L^{\prime}\rvert_{\infty}\leq\alpha(\alpha\lvert L\rvert_{\infty}+\beta+t^{*}_{k})+\beta\leq\alpha^{2}(\lvert L\rvert_{\infty}+2\beta+t^{*}_{k})$ .

Using the above reasoning for every player in the sequence, we have the displayed load profile $L_{n}$ at the end of the sequence has the property that $\lvert L_{n}\rvert_{\infty}\leq\alpha^{2n}(\lvert L_{0}\rvert_{\infty}+2n\beta+\sum_{k=1}^{n}t^{*}_{k})$ , where $L_{0}$ is the load profile shown to the first player. But $\lvert L_{0}\rvert_{\infty}$ is at most $\beta$ , since the true load on all machines is zero at that point.

Since the displayed makespan at the end of the sequence is at most $\alpha^{2n}(\beta+2n\beta+\sum_{k=1}^{n}t^{*}_{k})$ , hence the true makespan is at most $\alpha^{2n+1}(\beta+2n\beta+\sum_{k=1}^{n}t^{*}_{k})+\beta$ . Since $OPT\geq\sum_{k=1}^{n}t^{*}_{k}/m$ we have our result. ∎

A.2 Cut games

A cut game is defined by a graph, where every player is a node of the graph. Each of the $n$ players chooses one of the two colors, ‘red’ or ‘blue’, and the utility to a player is the number of her neighbors who do not have the same color as hers.

In sequential play, when a player has her turn to play, she is shown counts of the number of her neighbors who are colored ‘red’ and who are colored ‘blue’. We assume each player knows the total number of her neighbors in the graph exactly. With greedy strategies, each player chooses the color with fewer nodes when it is her turn to play. As was the case for machine scheduling, undominated strategies for cut games perform much worse than $OPT$ , even with perfect counters.

Lemma 5.

With perfect counters and undominated strategies, the competitive ratio against the optimal social welfare is at least $n$ .

Proof.

Consider the graph to be a long cycle with $2n$ nodes. For ease of analysis, number the nodes 0 through $2n-1$ with the node numbered $i$ have its neighbors $(i-1)\mod 2n$ and $(i+1)\mod 2n$ . The optimal social welfare is $4n$ obtained by coloring all even numbered nodes with red and the rest with blue.

Consider the sequence of nodes where nodes arrive in the increasing order $0$ through $2n-1$ . We claim that through a series of undominated strategy plays on part of each player, the coloring where node $2n-1$ is colored red and the rest colored blue is achievable. Note that this coloring gives a social welfare of 4.

We now prove our claim. It is an undominated strategy for node 0 to choose the color blue. Node 1 sees one of its neighbors colored blue and the other uncolored. It is an undominated strategy for node 1 to choose color blue as well. This continues and each node until node $2n-1$ is colored blue. Node $2n-1$ has both its neighbors colored blue, and so the only undominated strategy for her is to play red. ∎

Given the previous result, we focus our attention on greedy strategies. With greedy strategies and perfect counters, the competitive ratio is constant, shown by Leme et al. [12]. We show that, with privacy-preserving counters, it is possible to compare the social welfare of greedy to that of $OPT$ .

Theorem 10.

[12] With perfect counters and greedy strategies, the competitive ratio against the optimal social welfare is at most $2$ .

Proof.

Consider the choice made by player $t$ when it is her turn to play. Let $C_{t}$ be the number of neighbors of player $t$ that have adopted a color by the time it her turn to play. Notice that the total number of edges in the graph is $\sum_{t}C_{t}$ . Furthermore, the greedy strategy ensures that player $t$ gets value at least $C_{t}/2$ . Since the number of edges in the graph is an upper bound on the optimal social welfare, hence we have the greedy strategy achieving a competitive ratio of $2$ . ∎

Now, we compare the performance of greedy w.r.t. to approximate counters to $OPT$ .

Theorem 11.

With $(\alpha,\beta,\gamma)$ -counter vector and greedy strategies, with probability at least $1-\gamma$ , the social welfare is at least $\frac{OPT}{2\alpha^{2}}-\frac{2\beta}{\alpha}n$ .

Proof.

Let us analyze the play made by player $t$ when it is her turn to play. Let $R_{t}$ and $B_{t}$ be the true counts of red and blue neighbors of $t$ at that time, and without loss of generality let $R_{t}\geq B_{t}$ . Either the player chooses the blue color and this guarantees her utility of $C_{t}/2$ , where $C_{t}=R_{t}+B_{t}$ . On the other hand, if the player were to choose the color red, it must be the case that the displayed value of the blue counter is at least the displayed value of the red counter. For this to be true, it must be the case that $\alpha B_{t}+\beta\geq R_{t}/\alpha-\beta$ , and therefore $B_{t}\geq R_{t}/\alpha^{2}-2\beta/\alpha\geq C_{t}/(2\alpha^{2})-2\beta/\alpha$ . Hence, in either case, the player achieves utility of at least $C_{t}/(2\alpha^{2})-2\beta/\alpha$ .

Following the analysis used in the proof of Theorem 10, we have the result. ∎

A.3 Cost sharing games

A cost sharing game is defined as follows. $n$ players each have to choose one of the $m$ sets. There is an underlying bipartite graph between the players and the sets, and a player can choose only one among those sets that she is adjacent to (i.e., she shares an edge with). Moreover, every set $i$ has a cost $c_{i}$ and the cost to a player is the cost of the set she chooses divided by the number of players who chose that set i.e., each of the players who choose a particular set share its cost equally. Each player would like to minimize her cost; the social welfare is the sum of costs of the players, which is equal to the sum of the costs of the sets chosen by various players.

Leme et al. [12] prove that the sequential price of anarchy is $O(\log(n))$ . Our work uses counters to publicly display an estimate of the number of the players who have selected that set so far. With perfect counters, this estimate is always exact. Unfortunately, greedy strategies can perform poorly in this setting, even with exact counters.

Lemma 6.

With perfect counters and greedy strategies, the competitive ratio is $n$ .

Proof.

We first show that the competitive ratio is at most $n$ . Let $\widehat{s}_{i}$ be the set that $i$ should choose in the optimal allocation, and let $s_{i}$ she chose. Also, let $l(s_{i})$ be the number of player who chose set $s_{i}$ . Greedy strategy dictates that it must be the case that $c_{s_{i}}/l(s_{i})\leq c_{\widehat{s}_{i}}$ . Summing over all players $i$ , we have the total cost of the allocation produced by the mechanism is $\sum_{i=1}^{n}c_{s_{i}}/l(s_{i})\leq\sum_{i=1}^{n}c_{\widehat{s}_{i}}$ , and this is equal to $\sum_{j\in J}q(j)c_{j}$ , where $J$ is the collection of sets picked in the optimal allocation and $q_{j}$ is the number of players allocated to set $j$ . Since the optimal cost is $\sum_{j\in J}c_{j}$ and $q_{j}\leq n$ , we have the competitive ratio is at most $n$ .

We now show that the competitive ratio is at least $n$ . Consider the case where there is a public set $s$ that is adjacent to all the players and has cost $1+\epsilon$ (for any small $\epsilon>0$ ). In addition, there are $n$ private sets $s_{1},\cdots,s_{n}$ with set $s_{i}$ having cost 1 and adjacent only to player $i$ . In the sequential game play, with greedy strategies and perfect counters (indicating the number of players who have chosen a particular set so far in the game), each player will choose her private set since that will have cost 1 as opposed to $1+\epsilon$ for the public set. This gives a total cost of $n$ . The optimal solution is to pick the public set with a total cost of $1+\epsilon$ . ∎

In light of Lemma 6, the greedy strategy with respect to approximate counters should not perform well with respect to $OPT$ . However, we do show that there are instances in which greedy with respect to these approximate counters can be better than greedy with respect to perfect counters. The example we use is the same as in Lemma 6, and is also to the example showing the price of anarchy for cost-sharing is $\Omega(n)$ . Proposition 3 and the exponential improvement of the sequential price of anarchy over the simultaneous price of anarchy [12] suggest the instability of this equilibrium.

Proposition 3.

In certain instances of cost sharing with greedy strategies, the competitive ratio using privacy-preserving counters is better than using perfect counters.

Proof.

Consider the same instance as in Lemma 6. There is a public set that is adjacent to all the players and has cost $1+\epsilon$ . In addition, there is a private set for each player that is adjacent to only that player. Each private set has cost 1. The number of players is $n$ and the number of sets is $m=n+1$ .

Consider the following construction of the counter vector (here $p=1$ , $q=O(\log(n)\log(n^{2}m)/\epsilon)$ , $r=1/n$ and $c=8(p^{2}+2pq)$ ). For the initial sequence of $c$ players, for each player $i\in[c]$ , for each counter, a uniformly randomly chosen number in the range $[0,c]$ (drawn independently for each counter) is displayed. Starting with the $(c+1)$ st player, each counter in the counter vector displays the value according to $(p,q,r)$ –Tree-sum based construction (Lemma 2). It is easy to verify that the construction gives a $(\alpha,\beta,\gamma)$ counter vector for $\alpha=p$ , $\beta=c$ and $\gamma=r$ .

Let $P$ be the counter that corresponds to the public set, and $S_{i}$ be the counter for the $i^{th}$ private set in the counter vector. Initially, the true value of all the counters is 0. For the initial set of $c$ players, for each $i\in[c]$ , the probability that the displayed value of $P$ is greater than that of $S_{i}$ is $1/2$ (since for each player $i\in[c]$ , on each counter, a uniformly random number drawn independently from the range $[0,c]$ is displayed).

Hence, in the first $c$ players, the expected number of players for who the displayed value of $P$ is greater than the corresponding $S_{i}$ is $c/2$ , and under greedy strategy, all these players will choose the public set. Hence the expected true count of the $P$ at the end of the prefix of $c$ players is $c/2$ . Using a Chernoff bound, the probability the true count of $P$ after the first $c$ players is smaller than $c/4$ is at most $e^{(-c/16)}$ .

After the initial sequence of $c$ players, the counter values are displayed according to the $(p,q,r)$ -Tree based construction. By the error guarantees, it follows that if the true count for the public set is at least $p^{2}+2pq$ at the end of the initial $c$ -length sequence, then for the rest of the players, with probability $(1-r)$ , the displayed value of $P$ is always strictly greater than the displayed value of every $S_{i}$ (whose true value is at most 1 and so the displayed value is at most $p+q$ ). Since $c/4=2(p^{2}+2pq)$ , we can infer that with probability at least $(1-r-e^{(-c/16)})$ , all players after the initial sequence of length $c$ will choose the public set giving the total cost of at most $1+\epsilon+c$ . In contrast, with perfect counters, the total cost is always $n$ (Lemma 6). ∎

Appendix B Future Independent: Discrete Version

See 1

The proof of this Theorem follows from the connection between future-independent resource-sharing and online vertex-weighted matching, which we mention below.

Observation 2.

In the setting where $\|a_{i}\|_{1}=1$ for all $a_{i}\in A_{i}$ , for all $i$ , full-information, discrete resource-sharing reduces to online, vertex-weighted bipartite matching.

Proof.

Construct the following bipartite graph $G=(U,V,E)$ as an instance of online vertex-weighted matching from an instance of the future-independent resource sharing game. For each resource $r$ , create $n$ vertices in $V$ , one with weight $V_{r}(t)$ for each $t\in[n]$ . As players arrive online, they will correspond to vertices in $u_{i}\in U$ . For each $a_{i}\in A_{i}$ corresponding to a set of resources $S$ , $u_{i}$ is allowed to take any subset of $V$ with a single copy of each $r\in S$ . ∎

The proof of the social welfare is quite similar to the one-to-one, online vertex-weighted matching proof of [10], with the necessary extension for many-to-one matchings (losing a factor of $1/2$ in the process).

Proof of Theorem 1.

Consider any instance of $G=(U,V,E)$ , a vertex-weighted bipartite graph. Let $\mu$ be the optimal many-to-one matching, which can be applied to nodes in both $U$ and $V$ (where $u\in U$ has potentially multiple neighbors in $V$ ). Consider $\mu^{\prime}$ , the greedy many-to-one matching for a particular sequence of arrivals $\sigma$ .

Consider a particular $u\in U$ , and the time it arrives $\sigma(u)$ as $\mu^{\prime}$ progresses. If at least $1/2$ the value of $\mu(u)$ is available at that time, then $w(\mu^{\prime}(u))\geq\frac{1}{2}w(\mu(u))$ (since $u$ can be matched to any subset of $\mu(u)$ , by the downward closed assumption). If not, then $w(\mu^{\prime}(\mu(u)))\geq\frac{1}{2}w(\mu(u))$ (at least half the value was taken by others). Thus, we know that, for all $u$ ,

w(\mu^{\prime}(u))+w(\mu^{\prime}(\mu(u)))\geq\frac{1}{2}w(\mu(u))

summing up over all $u$ , we get

\displaystyle\sum_{u}w(\mu^{\prime}(u))+w(\mu^{\prime}(\mu(u)))=2w(\mu^{\prime})\geq\frac{1}{2}\sum_{u}w(\mu(u))=\frac{1}{2}w(\mu)

Rearranging shows that $w(\mu^{\prime})\geq\frac{1}{4}w(\mu)$ .

Finally, the utility to a player is clearly greatest when they are greedy, so that is a dominant strategy (thus implying any non-greedy strategy is dominated). ∎

B.1 Greedy play with more accurate estimates

Observation 3.

Suppose that $\mathcal{M}$ is a $(\alpha,\beta,\gamma)$ -underestimating counter vector, giving estimates $y_{i,r}$ . Furthermore, assume each player $i$ is playing greedily with respect to a revised estimate $z_{i,r}$ such that, for each $r,i,$ and value of $z_{i}{r}$ is always in the range $[y_{i,r},x_{i,r}]$ . Then, for $g$ , a discrete resource-sharing game, with probability $1-\gamma$ , the ratio of the optimal to the achieved social welfare is $O(\alpha\beta)$ .

Proof.

The proof follows from the proof of Theorem 4, along with the following observation. Since $z_{i,r}$ ’s is deterministically more accurate than the counters, we have for each $i$ that the value gained by greedily choosing according to the estimates $z_{i,r}$ is at least as much as the value gained by greedily choosing using $y_{i,r}$ . Therefore, summing over all the players, the achieved social welfare is at least as much as it would be if everyone had played greedily according to $y_{i,r}$ . ∎

Observation 4.

There exists a resource-sharing game $g$ , such that if the players play greedily according to estimates $z_{i,r}$ that are more accurate than the displayed value only in expectation – specifically for each $r,i,$ and value of $x_{i,r}$ , $\mathbb{P}[z_{i,r}<x_{i,r}]\geq 1/2$ and also $\mathbb{E}\left[|z_{i,r}-x_{i,r}|\right]=1$ , then the ratio of the optimal to the achieved social welfare can be as bad as $\Omega\left(\sqrt{n}\right)$ .

Proof.

Let there be $n+\sqrt{n}$ resources, with resources $r*_{1,\ldots,\sqrt{n}}$ having $V_{r*_{f}}(0)=H$ , $V_{r*_{f}}(t)=0$ for all $t>0$ , and resource $r_{i}$ such that $V_{r_{i}}(t)=H-\epsilon$ for all $t$ . Player $i$ has access to all resources $r*_{f}$ and $r_{i}$ . Then, $OPT=H\sqrt{n}+(H-\epsilon)(n-\sqrt{n})=Hn-(n-\sqrt{n})\epsilon$ .

Consider the counter vector which is exactly correct with probability $1-\frac{1}{\sqrt{n}}$ and undercounts by $\sqrt{n}$ with probability $\frac{1}{\sqrt{n}}$ (note that the expected error is just $1$ and it undercounts with probability 1). Then, greedy behavior with respect to this counter will (in expectation) have $\sqrt{n}$ players choose $r*_{f}$ for each $f$ , achieving welfare $\sqrt{n}H$ . Thus, the competitive ratio is $\Omega(\sqrt{n})$ as $\epsilon\to 0$ , as desired. ∎

B.2 Undominated strategic play with Empty Counters: Lower bounds

See 2

Proof.

Let $g$ be the following game. For each player $i$ , there is a resource $r_{i}$ such that $v_{r_{i}}(1)=H$ but $v_{r_{i}}(>1)=0$ . Furthermore, let there be some other resource $r$ such that $v_{r}(1)=1$ . Let $A_{i}$ contain 2 allowable actions: selecting $r_{i}$ and selecting $r$ .

OPT in this setting would have each player select $r_{i}$ , which has $SW(OPT)=nH$ . On the other hand, we claim it is undominated for each player to select $r$ instead (call this joint action $a$ ). If each player were to have a “twin”, then $r_{i}$ could have already been selected by another player so that $i$ would get more utility from $r$ than $r_{i}$ . Then, this undominated strategy $a$ has $SW(a)=n$ . Thus, we have a game $g$ for which

CR_{\textsc{Undom}}(g)\geq\frac{nH}{n}=H

which, as $H\to\infty$ is unbounded. ∎

The negative result above isn’t particularly surprising: if there is some coordination to be done, but there is no coordinator and no information about the target, all is lost. On the other hand, our positive result for undominated strategies (Theorem 7) in the case of private information relies on a very particular rate of decay of the resources’ value. Theorem 3 show that, even under this stylized assumption where all resources’ values shrink slowly, a total lack of information can lead to very poor behaviour in undominated strategies.

See 3

Proof of Theorem 3.

For each player $i$ , let $r_{i}$ be a resource where $v_{r_{i}}(1)=n$ (note that this uniquely determines $v_{i}(c)$ for all $c$ ). Let there be another resource $r$ such that $v_{r}(1)=1$ . Let each $A_{i}$ contain all resources. Since $\frac{v_{r_{i}}(1)}{n}=1$ , it is not dominated for player $i$ to select $r$ . Let $a$ denote the joint strategy where each player selects resource $r$ . Thus the social welfare attained by this strategy profile is $O(\log(n))$ , where as the optimal social welfare is $n^{2}$ , implying that $CR_{\textsc{Undom}}\geq\Omega(\frac{n^{2}}{\log(n)})$ . ∎

B.3 Omitted proofs for Undominated strategies with Privacy-preserving counters

Proof of Theorem 7.

Consider the optimal allocation and let $r_{i}$ and $z_{i}$ denote that the $z_{i}^{th}$ copy of resource $r_{i}$ got allocated to player $i$ under the optimal allocation. Now consider any run of the game under undominated strategic play and based on the run, partition all the players into two groups. Group $A$ consists of players $i$ such that $x_{i,r_{i}}\leq z_{i}$ (i.e., the copy (or a more valuable copy) of the resource that was allocated to player $i$ was present when the player arrived) and group $B$ consists of all other players.

For the player in group $B$ , the copy of the resource that they received in the optimal allocation was already allocated by the time they arrived in the run of the undominated strategic play. Hence, the total social welfare achieved by the undominated strategic play is at least as much the welfare achieved by group $B$ player under optimal allocation.

Now consider any player $i$ in group $A$ . We show that the resource picked by player $i$ under undominated strategic play gets her a reasonable fraction of the value she would have received under optimal allocation. For any resource $r$ , given the displayed counter value of $y_{i,r}$ , by the guarantees of the $(\alpha,\beta)$ -accuracy guarantee of the counters, we directly argue about the possible range of the consistent beliefs or estimates $\widehat{x_{i,r}}$ by which player $i$ can make her choice.

Specifically, by the bounds on $(\alpha,\beta)$ -counters, for a given true value $x$ , it must be the case that all announcements $y_{i,r}$ satisfy:

\alpha x_{i,r}+\beta\geq y_{i,r}\geq\frac{1}{\alpha}x_{i,r}-\beta

Rearranging, we have $y_{i,r}\in[\frac{1}{\alpha}x_{i,r}-\beta,\alpha x_{i,r}+\beta]$ . Suppose these bounds are realized; we wish to upper and lower bound $\widehat{x_{i,r}}$ as a function of these announcement values. By the quality of the announcement, we have that $\alpha\widehat{x_{i,r}}+\beta\geq y_{i,r}\geq\frac{1}{\alpha}x_{i,r}-\beta$ .

We can similarly upper bound $\widehat{x_{i,r}}$ , e.g. $\alpha x_{i,r}+\beta\geq y_{i,r}\geq\frac{1}{\alpha}\widehat{x_{i,r}}-\beta$ , which, by the fact that the true count is at least 0, implies $\widehat{x_{i,r}}\in[\max\{0,\frac{x_{i,r}}{\alpha^{2}}-\frac{2\beta}{\alpha}\},\alpha^{2}x_{i,r}+2\alpha\beta].$

Now, suppose player $i$ chose resource $r^{\prime}$ which was undominated and not $r_{i}$ which he received in the optimal allocation. Since resource $r^{\prime}$ is undominated:

\displaystyle V_{r^{\prime}}(\widehat{x_{i,r^{\prime}}})\geq V_{r_{i}}(\widehat{x_{i,r_{i}}})

(1)

We have

\displaystyle V_{r^{\prime}}(\widehat{x_{i,r^{\prime}}})\leq V_{r^{\prime}}(\max\{0,\frac{x_{i,r^{\prime}}}{\alpha^{2}})-\frac{2\beta}{\alpha}\})\leq\psi(\alpha,\beta)V_{r^{\prime}}(x_{i,r^{\prime}})

(2)

where the first inequality came from the lower bound on the counter, and the fact that the valuations are decreasing, and the second from the assumption about $V_{r}$ on $x$ and its lower bound. Similarly, we know for each $r$ that

\displaystyle V_{r_{i}}(\widehat{x_{i,r_{i}}})\geq V_{r_{i}}(\alpha^{2}x_{i,r_{i}}+2\alpha\beta)\geq\frac{V_{r_{i}}(x_{i,r_{i}})}{\phi(\alpha,\beta)}

(3)

Combining the three equations above, we have the actual value received by the player $i$ on choosing resource $r^{\prime}$ , $V_{r^{\prime}}(x_{i,r^{\prime}})$ is at least $\frac{1}{\psi(\alpha,\beta)\phi(\alpha,\beta)}$ fraction of the value $V_{r_{i}}(x_{i,r_{i}})$ that he would receive under the optimal allocation. Therefore, by virtue of partition of the players in groups $A$ and $B$ , we have that social welfare achieved under undominated strategic play is at least $\frac{1}{1+\psi(\alpha,\beta)\phi(\alpha,\beta)}$ fraction of the optimal social welfare. ∎

Appendix C Resource Sharing: Continuous version

In this section, we allow investments in resources to be non-discrete. The utility of player $i$ in the continuous model is the following:

u_{i}(a_{1},\ldots,a_{n})=\sum_{r=1}^{m}\int_{x_{i,r}}^{x_{i,r}+a_{i,r}}v_{r}(t)dt,

where $x_{i,r}=\sum_{i^{\prime}=1}^{i-1}a_{i^{\prime},r}$ is the amount already invested in resource $r$ by earlier players.

In this setting, in order to prove a theorem analogous to Theorem 4 in the discrete setting, we need an analogue to Lemma 1 that holds in the full-information continuous setting. We no longer have the tight connection between our setting and matching; nonetheless, the fact that the greedy strategy is a $4$ -approximation to $OPT$ continues to hold.

Lemma 7.

The greedy strategy for many-to-one online, continuous, resource-weighted “matching”, where players arrive online and have tuples of allowable volumes of resources, has a competitive ratio of $\frac{1}{4}$ .

Proof Sketch.

The proof is identical to the proof of Lemma 1, with the exception that we no longer want matchings $\mu,\mu^{\prime}$ but rather correspondences between continuous regions of $v^{\prime}_{r}$ . See Figure 1 for a visual proof sketch. ∎

Figure 1: Suppose the blue regions are those selected by the players who got those regions in OPT, and the red regions are those selected by some other player. Then, if some greedy player(s) have taken at least half of the value of the optimal regions for another player, at least that much utility has been gained by the greedy players. If not, half the value is still available for the player at hand.

With Lemma 7, following analysis similar to Theorem 4, we have the following.

Theorem 12.

Suppose that $\mathcal{M}$ is an $(\alpha,\beta,\gamma)$ -underestimating counter vector. Then, for any continuous, future-independent resource-sharing game $g$ , $CR_{\textsc{Greedy}}(\mathcal{M},g)=O(\alpha\beta)$ .

Appendix D Resource Sharing with Future-Dependent utilities

The second model of utility we consider is one where the benefit of choosing a resource for a player depends not only the actions of the past players but also on the actions taken by future players. Specifically, all the players who selected a given resource incur the same benefit regardless of the order in which they made the choice. The utility of player $i$ ,

u_{i}(a_{1},\ldots,a_{n})=V_{r}(x_{r}),

where $r$ is the resource chosen by player $i$ and $x_{r}=\sum_{i^{\prime}=1}^{n}a_{i^{\prime},r}$ is the total utilization of resource $r$ by all players. As a warm-up, we start with the special case of market sharing in the section below, and then move to the case of more general value curves.

D.1 Market sharing

Market sharing is the special case of $V_{r}(x_{r})=c/x_{r}$ for all $x_{r}\geq 1$ . We show that with greedy play and private counters, it is possible to achieve a logarithmic factor approximation to the social welfare, while with undominated strategies and perfect counters, one cannot hope to achieve an approximation that is linear in the number of the players.

Goemans et al. [7] showed that for market-sharing games, the competitive ratio of $\alpha$ -approximate greedy play is at most $O(\alpha\log(n))$ . Using analysis similar to theirs, we have the following result.

Corollary 3.

With $(\alpha,\beta,\gamma)$ -counter vector and greedy play, with probability at least $1-\gamma$ , the welfare achieved is at least $(OPT-2\beta\alpha n)/O((1+\alpha^{2})\log(n))$ .

For undominated strategies, we have the following result.

Lemma 8.

With perfect counters and undominated strategic play, there are games for which the welfare achieved is at most $OPT/(n\log(n))$ .

Proof.

Here is an example with $n$ players. Consider the case where for every $i\geq 1$ , player $i$ is interested in resource 0 and resource $i$ . For every $i\geq 1$ , the total value of resource $i$ is $(n-i+1)(1-\epsilon)/i$ (for some small $\epsilon>0$ ). The value of resource 0 is 1.

We claim that there is an undominated strategy game play where every player chooses resource 0 giving a social welfare of 1, whereas the optimal welfare is achieved by assigning player $i$ resource $i$ giving a total welfare of $n(\log(n)-1)(1-\epsilon)$ .

Here is such an undominated strategy profile: for each $i$ , player $i$ believes that every player after her is only interested in resource $i$ . With this belief, it is easy to see that choosing resource 0 is an undominated strategy for every player. ∎

D.2 General value curves

In this general setting, we will be interested in value curves that do not decrease too quickly. Furthermore, we study only the greedy strategy since we have already seen that undominated strategy does not perform well even for simple curves (Lemma 8).

Definition 4 ( $(w,l)$ -shallow value curve).

A value curve $V_{r}$ is $(w,l)$ -shallow if for all $x\leq l$ , it is the case that $V_{r}(x)\geq\frac{\sum_{t=0}^{x}V_{r}(t)}{wx}$ .

The definition of $(w,l)$ -shallow value curve says that the actual payoff all players get from the resource being utilized with $x$ weight is not too much smaller than the integral of $V_{r}$ from $0$ to $x$ .

In the following result, we show that this restriction on the rate of decay of the value curves is necessary to say anything nontrivial about the performance of the greedy strategy.

Lemma 9.

Even with perfect counters, there exist sequential resource-sharing games $g$ , where each resource $r$ ’s value curve $V_{r}$ is $(w,n)-shallow$ , such that in the future-dependent setting, $CR_{\textsc{Greedy}}(\mathcal{M}_{Full},g)\geq 2w$ .

Proof.

Consider two players and two resources $r,r^{\prime}$ . Let $r$ have a value curve which is a step function, with $v_{r}(0)=w$ , $v_{r}(1)=\frac{1}{2}$ and $v_{r^{\prime}}(0)=w-\epsilon$ . Suppose player one has access to both resources and player two has only resource $r$ as an option. Then, player one will choose $r$ according to greedy, and player two will always select $r$ . The social welfare will be $SW(\textsc{Greedy})=1$ , whereas $OPT$ is for player $1$ to take $r^{\prime}$ and will have $SW(OPT)=2w-\epsilon$ . As $\epsilon\to 0$ , this ratio approaches $2w$ . ∎

Thus, as $w\to\infty$ , the competitive ratio of the greedy strategy is unbounded. Fortunately, the competitive ratio cannot be worse than this, for fixed $w$ , as we show in the theorem below.

Theorem 13.

Suppose, for a sequential resource-sharing game $g$ , each resource $r$ ’s value curve $v_{r}$ is $(w,n)-shallow$ . Then, in the in the future-dependent setting, $CR_{\textsc{Greedy}}(\mathcal{M},g)=O(w\alpha\beta)$ for an $(\alpha,\beta)$ -counter $\mathcal{M}$ .

Proof Sketch.

According to the greedy strategy, player $i$ chooses the resource in $A_{i}$ that maximizes $V_{r}(x_{i,r}+1)$ , and we say $V_{r}(x_{i,r}+1)$ is her perceived value if $r$ is the resource she chose. Terming the sum of the perceived values of all players as the perceived social welfare, $PSW(\textsc{Greedy})$ , we have

\begin{split}PSW(\textsc{Greedy})=&\sum_{i\in[n]}\sum_{r}a_{i,r}V_{r}(x_{i,r}+1)dx=\sum_{r}\sum_{x=0}^{x_{n,r}+a_{n,r}}V_{r}(x)\\ \leq&\sum_{r}w\left(x_{n,r}+a_{n,r}\right)V_{r}(x_{n,r}+a_{n,r})=w\,SW(\textsc{Greedy})\end{split}

(4)

where the last inequality comes from our assumption about the value curves all being $(w,n)-shallow$ .

The final part of the argument must show that the actual welfare from greedy play with respect to the counters is well-approximated by the perceived welfare with respect to the true counts. Since the counters are accurate within some quantity $\leq n$ , this is the case. Following an analysis similar to that of Theorem 4,we have our result. ∎

Appendix E Analysis of Private Counters

Proof of Lemma 2.

We assume the reader is familiar with the TreeSum mechanism. The privacy of this construction follows the same argument as for the original constructions. One can view $m$ independent copies of the TreeSum protocol as a single protocol where the Laplace mechanism is used to release the entire vector of partial sums. Because the $\ell_{1}$ -sensitivity of each partial sum is 1 (since $\|a_{t}\|\leq 1$ ), the amount of Laplace noise (per entry) needed to release the $m$ -dimensional vector partial sums case is the same as for a dimensional $1$ -dimensional counter.

To see why the approximation claims holds, we can apply Lemma 2.8 from [3] (a tail bound for sums of independent Laplace random variables) with $b_{1}=\cdots=b_{\log n}={\log n}/\epsilon$ , error probability $\delta=\gamma/mn$ , $\nu=\frac{(\log n)\sqrt{\log(1/\delta)}}{\epsilon}$ and $\lambda=\frac{(\log n)(\log(1/\delta)}{\epsilon}$ , we get that each individual counter estimate $s_{t}(j)$ has additive error $O(\frac{(\log n)(\log(nm/\gamma))}{\epsilon})$ with probability at least $1-\gamma/(mn)$ . Thus, all $n\cdot m$ estimates satisfy the bound simultaneously with probability at least $1-\gamma$ . ∎

Proof of Lemma 3.

We begin with the proof of privacy. The first phase of the protocol is $\epsilon/2$ -differentially private because it is an instance of the “sparse vector” technique of Hardt and Rothblum [8] (see also [17, Lecture 20] for a self-contained exposition). The second phase of the protocol is $\epsilon/2$ -differentially private by the privacy of TreeSum. Since differential privacy composes, the scheme as a whole is $\epsilon$ -differentially private. Note that since we are proving $(\epsilon,0)$ -differential privacy, it suffices to consider nonadaptive streams; the adaptive privacy definition then follows [4].

We turn to proving the approximation guarantee. Note that the each of the Laplace noise variables added in phase 1 of the algorithm (to compute $\tilde{x_{t,r}}$ and $\tau_{j}$ ) uses parameter $2/\epsilon^{\prime}$ . Taking a union bound over the $mn$ possible times that such noise is added, we see that with probability at least $1-\gamma/2$ , each of these random variables has absolute value at most $O(\frac{\log(mn/\gamma)}{\epsilon^{\prime}}$ . Since $\frac{2}{\epsilon^{\prime}}=O(\frac{mk}{\epsilon})$ and $k=O(\log\log(\frac{nm}{\gamma})+\log\frac{1}{\epsilon})$ , we get that each of these noise variables has absolute value $\tilde{O}_{\alpha}(\frac{m\log(mn/\gamma)}{\epsilon})$ with probability all but $\gamma/2$ . We denote this bound $E_{1}$ .

Thus, for each counter, the $i$ -th flag is raised no earlier than when the value of the counter first exceeds $\alpha^{i}(\log n)-E_{1}$ , and no later than when the counter first exceeds $\alpha^{i}(\log n)+E_{1}$ . The very first flag might be raised when counter has value 0. In that case, the additive error of the estimate is $\log n$ , which is less than $E_{1}$ . Hence, he mechanism’s estimates during the first phase provide an $(\alpha,E_{1},\gamma/2)$ -approximation (as desired).

The flag that causes the algorithm to enter the second phase is supposed to be raised when the counter takes the value $A:=\alpha^{k}(\log n)\geq\frac{\alpha}{\alpha-1}\cdot C_{tree}\cdot\frac{\log(nm/\gamma)}{\epsilon}$ ; in fact, the counter could be as small as $A-E_{1}$ . After that point, the additive error is due to the TreeSum protocol and is at most $B:=C_{tree}\cdot\log(n)\cdot\log(nm/\gamma)/\epsilon$ (with probability at least $1-\gamma/2$ ) by Lemma 2. The reported value $y_{i,r}$ thus satisfies

y_{i,r}\geq x_{i,r}-B=\frac{1}{\alpha}x_{i,r}+\underbrace{(1-\frac{1}{\alpha})x_{i,r}-B}_{\text{residual error}}\,.

Since $x_{i,r}\geq A-E_{1}$ , the “residual error” in the equation above is at least $(1-\frac{1}{\alpha})(A-E_{1})-B=-(1-\frac{1}{\alpha})E_{1}\geq-E_{1}$ . Thus, the second phase of the algorithm also provides $(\alpha,E_{1},\gamma/2)$ -approximation. With probability $1-\gamma$ , both phases jointly provide a $(\alpha,E_{1},\gamma)$ -approximation, as desired. ∎

Privacy-Preserving Public Information for Sequential Games

Abstract

1 Introduction

1.1 Game Model

Illustrative Example

1.2 Information Model

1.3 Players’ Behavior

1.4 Statement of Main Results

1.5 Related Work

2 Privacy-preserving public counters

Definition 1.

Definition 2 ((α,β,γ)(\alpha,\beta,\gamma)-accurate counter vector).

Definition 3 ((α,β,γ)(\alpha,\beta,\gamma)-accurate underestimator counter vector).

Observation 1.

Proof.

3 Resource Sharing

3.1 Perfect counters and empty counters

Theorem 1.

Theorem 2.

Theorem 3.

3.2 Privacy-preserving public counters and Greedy Strategy

Theorem 4.

Lemma 1.

Proof.

Proof of Theorem 4.

Theorem 5.

Proof.

3.3 Privacy-preserving public counters and Undominated strategies

Theorem 6.

Proof.

Theorem 7.

Corollary 1.

4 Private Counter Vectors with Lower Errors for Small Values

Lemma 2.

Lemma 3.

Enforcing Additional Guarantees

Proposition 1.

Proposition 2 ([4]).

Corollary 2.

5 Extensions

6 Discussion and Open Problems

References

Appendix A Other games

A.1 Unrelated Machine scheduling games

Lemma 4.

Proof.

Theorem 8.

Theorem 9.

Proof of Theorem 9.

A.2 Cut games

Lemma 5.

Proof.

Theorem 10.

Proof.

Theorem 11.

Proof.

A.3 Cost sharing games

Lemma 6.

Proof.

Proposition 3.

Proof.

Appendix B Future Independent: Discrete Version

Observation 2.

Proof.

Proof of Theorem 1.

B.1 Greedy play with more accurate estimates

Observation 3.

Proof.

Observation 4.

Proof.

B.2 Undominated strategic play with Empty Counters: Lower bounds

Proof.

Proof of Theorem 3.

B.3 Omitted proofs for Undominated strategies with Privacy-preserving counters

Proof of Theorem 7.

Appendix C Resource Sharing: Continuous version

Lemma 7.

Proof Sketch.

Theorem 12.

Appendix D Resource Sharing with Future-Dependent utilities

Definition 2 ( $(\alpha,\beta,\gamma)$ -accurate counter vector).

Definition 3 ( $(\alpha,\beta,\gamma)$ -accurate underestimator counter vector).

Definition 4 ( $(w,l)$ -shallow value curve).