On the Core of Dynamic Cooperative Games

Dynamic cooperative games have been studied in a few versions. Most of the studies, as we do, concentrate on the core. Oviedo (2000) studies the core of a finitely repeated discounted cooperative game where the stage game does not vary over time and no dynamic consideration is involved.

Kranich et al. (2005) consider a finite horizon of predetermined games. They study three different core concepts. The classical core assumes that coalitions planning to split off do so right at the beginning. This concept does not depend on the temporal structure of the game: the classical core coincides with the core of an induced static game. The strong sequential core, on the other hand, allows for deviations of coalitions at any stage of the game, but once a coalition deviates at some point, it must keep doing so from that time on. In the above two concepts deviations are not required to be credible, i.e., they could be blocked by some sub-coalition in the future. The weak sequential core is robust against credible coalitional deviations. The latter means that deviations are immune to deviations of sub-coalitions. The sub-coalition deviations can be themselves non-credible. Habis and Herings (2010) provide a correction of the above definition of weak sequential core.

Predtetchinski (2007) deals with infinite-horizon stationary cooperative games, where at each moment the game is in one of a finite number of states, that determines which instantaneous game is played at that moment. The states evolves according to an exogenous Markov chain and it does not depend on past allocations. The author considers the classical core and a version of the strong sequential core, and provides conditions for nonemptyness of the strong sequential core. Hellman (2008) focuses on the bargaining set of dynamic cooperative games, where the sequence of stage TU-games is exogenously specified.

Related results can be found in Gale (1978), where a concept of sequential core is defined and is used to model lack of trust in a two-period economy. In this model coalitions are allowed to deviate in the second period. Becker and Chakrabarti (1995) consider infinite horizon capital allocation models and define recursive core allocations, the ones where no coalition can improve upon its consumption stream at any time given its accumulation of assets up to that period.

Koutsougeras (1998) introduces the notion of two-stage core, that takes into account the possibility of temporary cooperation. Within each coalition agents make future trades only if they are enforceable, i.e., a coalition may have a limited horizon. Moreover a coalition blocks at some point in time only if it can secure improvements for its members in any possible consequence of a deviation. Predtetchinski et al. (2002, 2006) use the concepts of strong and weak sequential core in the context of two-period economies. Predtetchinski et al. (2004) apply the concept of strong sequential core to a stationary exchange economy.

Petrosjan (1977, 1993) deals with a cooperative game induced by a (non-cooperative) differential game and Petrosjan and Zaccour (2003) study the problem of allocation over time of total cost incurred by countries in a cooperative game of pollution reduction and compute the Shapley value of this game. These papers are not about dynamic cooperative games but about a cooperative game induced by non-cooperative game played over time.

There exists a whole literature on coalition formation, where stability of coalitions is considered under different aspects (see, e.g., Ray, 1989; Chwe, 1994; Xue, 1998; Ray and Vohra, 1999; Konishi and Ray, 2003; Diamantoudi and Xue, 2007; Ray, 2007, and references therein). Typically this literature considers strategically richer models than the one examined in this paper, so it is closer in spirit to noncooperative game theory. For instance the model considered by Konishi and Ray (2003), which describes coalition formation as a truly dynamical process, considers a state space, beliefs, a probabilistic structure, and equilibrium concepts.

Our notion of credible core can be related to the papers by Bernheim et al. (1987); Bernheim and Whinston (1987) on coalition-proof Nash equilibria.

In some of our results we resort to the concept of $\varepsilon$-core. This was introduced in Shapley and Shubik (1966) to analyze situations where the core is empty. It has been employed in different contexts by Wooders (1983); Shubik and Wooders (1983a, b); Wooders and Zame (1984); Kovalenkov and Wooders (2001a, b, 2003, 2005), among others. In some of these papers a parametrized collection of cooperative games is considered and an approximate core is computed, where the goodness of the approximation depends on the parameters of the game. In particular, given the parameters $\pi$ describing a collection of games and given a lower bound $n_{0}$ on the number of players in each game in the collection, Kovalenkov and Wooders (2001b) obtain a bound $\varepsilon(\pi,n_{0})$ so that, for any $\varepsilon\geq\varepsilon(\pi,n_{0})$, all games in the collection with at least $n_{0}$ players have nonempty $\varepsilon$-cores. Some of our results have a similar flavor, except that for us the lower bound on the $\varepsilon$ is zero and the quantity that guarantees the existence of the $\varepsilon$-core is the discount factor, rather than the number of players.

The paper is organized as follows. In Section 2 we give a motivating example based on the classical market games of Shapley and Shubik. The model is introduced in 3 and the two first types of core solutions are given in Section 4. Section 5 provides characterizations of the non-emptyness of the $\varepsilon$-core when the discount factor is sufficiently large. The credible core is discussed in Section 6 and the paper ends with a section devoted to a few final remarks.

Let $N=\{1,\dots,n\}$ be the set of players. For any coalition $S\subseteq N$ consider a function $v^{S}:2^{S}\to\mathbb{R}_{+}$ with $v^{S}(\varnothing)=0$. The function $v^{S}$ is called characteristic function defined over $S$ with the set $S$ being the grand coalition of $v^{S}$. An allocation of $v^{S}$ is a vector $x^{S}\in\mathbb{R}^{S}$ that satisfies $\sum_{i\in S}x^{S}(i)=v^{S}(S)$ and $x^{S}(i)\geq B$, where $B$ is a uniform lower bound over all allocations. The reason for this lower bound is primarily technical: with this lower bound the set of allocations becomes compact. If we take $B=0$, no inter-temporal loans are allowed, whereas, when $B<0$ a player can get less than her individually rational level at a certain stage, but then she will be compensated in the future.

At any stage $t$ a cooperative games over a grand coalition $S$ is played. Both the game and the grand coalition depend on the history up to that stage. The players of the grand coalition $S$ are getting at time $t$ an allocation of the game actually being played. The cooperative game of the subsequent period depends on the current allocation.

Formally, a dynamic cooperative game is played over a discrete set of periods. The evolution of the system depends on the initial game, the allocation at every period and the dynamic $V$, specified below. At stage $1$ any coalition decides whether to split off or not. If no coalition splits off, then the initial cooperative game $v_{1}^{N}$ is played, and the allocation is $x^{N}_{1}$. If $S$ splits off, then the initial cooperative game is $v_{1}^{S}$ and the allocation is $x^{S}_{1}$. Note that the initial game is predetermined and is beyond the control of the players, unless a sub-coalition wishes to split off.

Like in a static cooperative game we will use (different versions of) the core to determine whether the grand coalition $N$ is stable (in different senses), namely no sub-coalition $S$ has an incentive to split. In case a coalition $S$ can advantageously split, the grand coalition $N$ is not stable, which makes the core empty. That is why we do not need to specify what payoffs players in $N\setminus S$ get or whether some of them want to form a coalition of their own.

From period 2 on, the evolution is governed by $V$. The state of the system is a pair $(S;x^{S})$, where $S$ is a coalition and $x^{S}$ is an allocation of $S$. As long as no coalition splits off, the state is of the form $(N;x^{N})$; once a coalition $S\subseteqq N$ deviates, the system turns to a state of the type $(S;x^{S})$, and $S$ remains fixed from that stage on forever.

When the state at time $t$ is $(N;x^{N})$, unless a deviation of $S$ occurs, the game played at time $t+1$ is $V(N;x^{N})$, whose grand coalition is $N$. At this time the allocation is $x^{N}_{t+1}$. However, if $S$ deviates, the game played at time $t+1$ is $V(S;x^{N}_{S})$, where $x^{N}_{S}$ is the allocation induced by $x^{N}$ to coalition $S$. At time $t+1$ the allocation is $x^{S}_{t+1}$ and the subsequent game is $V(S;x^{S}_{t+1})$ whose grand coalition is $S$.

For the sake of simplicity we assume a Markovian structure of the game, where the stage game played at time $t$ depends on the allocation at time $t-1$. More complicated dynamics could be considered, for instance the game at time $t$ could depend on the whole past history. An interesting intermediate case is the one where the game depends on some unidimensional function of the history, for instance on the sum of the past allocations. Think for instance of a model of dynamic public good provision, where every player contributes to a public good, whose level at time $t$ depends on the (discounted) sum of past contributions.

A few remarks regarding our modeling choices are in place. Up to Section 6, we assume that once a coalition deviates, it remains the grand coalition forever and no further splitting off of sub-coalitions will take place. This restriction corresponds to the first two types of core solutions, that are concerned with long-term plans that prevent these kind of deviations. When dealing with the third type of core solution we lift this restriction. A deviating coalition is not protected against coups of its sub-coalitions. This is the reason why a threat of a coalition to deviate is rendered credible only if it is immunized against further split offs of its sub-coalitions.

We also assume here that once a coalition deviates, there will be no way to restore a full cooperation and to rebuild the grand coalition $N$. Such a possibility requires a much more complicated dynamic that would depend also on past allocations of non-deviating members. In this paper we decide to keep matters as simple as possible. This distinguishes our model from the literature on coalition formation that we mentioned in the Introduction.

In our model a coalition $S$ can deviate prior to time $1$ and play the game $v_{1}^{S}$. We could as well assume that at time $1$ for any $S$, $V(N,x^{N}_{0})(S)=V(S,x^{N}_{0})(S)$, and it will make no difference in the results. Note however, that in the motivating example, when the grand coalition is $N$, the worth of $S$ is typically different from its worth when the grand coalition is $S$.

Our model refers to the deviating coalition, but it ignores the rest of the players. In principle, the complement of a deviating coalition could be treated just like the deviating coalition itself. The continuation game of the complement depends on its historical allocations. However, we chose not to specify it for two reasons. First, our study focuses on two main aspects, fairness and stability. Whether or not a coalition is treated equitably does not depend on what happens to its complement. The same applies to the willingness of a coalition to split off: it is not affected by the its complement.

The second reason is that we study core solutions and characterize the games whose core is non-empty. In these games no deviation will occur and in any case no coalition will be left alone without its complement.

We assume throughout that all players have the same discount factor $0<\delta<1$. Suppose that player $i$’s payoff at time $t$ is $x_{t}(i)$. Her present normalized payoff is

We define for every $T\subseteq N$

Therefore ${x}_{*}(T,\delta)$ is the sum of all individual allocations of $T$’s members.

Define a new characteristic function on $N$ as follows.

where the maximum is taken over all feasible histories of $S$-allocations: $x^{S}_{1},x^{S}_{2},\dots$ and $x_{t}(i)=x^{S}_{t}(i)$.

The dynamic game with discount factor $\delta$ will be denoted by $(V,\delta)$.

There are two justifications for the definition of core in the classical model of one-shot cooperative game. The first is fairness and the second is stability. An allocation is in the core if any coalition obtains at least its worth. Behind this justification lies an assumptions that a central planner has a full control on what the players get, and once she makes up her mind regarding the split of the cake, the players have no way to protest.

This kind of reasoning leads us to define what we call the ‘fair core’ first. A sequence of allocations $x^{N}_{1},x^{N}_{2},\dots$ is in the fair core if fore every coalition $S$ the present value of the shares of $S$ exceeds the present value of the worths of $S$. It is assumed that coalition $S$ can do nothing about its future shares, which gives the central planner the freedom to choose allocations without paying attention to semi-strategic considerations like stability.

The definition of fair core is concerned solely with the following consideration: it is fair to give coalition $S$ allocations whose present value is no less than what their present worth is.

If the core of the stage game at time $t$ is nonempty, for every $t$, then the fair core of $(V,\delta)$ is nonempty.

That is, $x^{N}_{1},x^{N}_{2},\dots$ is efficient if the present value of the grand coalition’s share is the maximum available. Recall that the actual game is history dependent. While any of the stage allocations could be locally efficient, it might reduce the size of the cake in subsequent periods and thereby might hamper efficiency.

In the classical one-shot game the definition of allocation contains the requirement of efficiency. This is not the case in the dynamic game. The fair core is not necessarily efficient, as demonstrated by the following example.

In the special case where the worth of the grand coalition is constant, that is when the worth of the grand coalition in all stage games do not depend on the history of allocation nor on the time, the fair core and the efficient fair core coincide.

The definition of the fair core does not make use of the entire structure of the dynamic game. It uses only states of the type $(N;x^{N})$ and not of the type $(S;x^{S})$, where $S\subsetneqq N$.

The second justification of the core involves stability considerations. An under-treated coalition might deviate, create its own game, and improve its position by reallocating its endowment. When a coalition $S$ threatens to deviate, it shifts the system to a state of the form $(S;x^{S})$ and the dynamic then is governed by $V(S;\cdot)$. This is why referring just to states of the type $(N;x^{N})$ is insufficient and there is a need to refer to states of the form $(S;x^{S})$ for every coalition $S$.

Let $x^{N}_{1},x^{N}_{2},\dots$ be a sequence of allocations. Define

The number $x^{h}_{*}(i,\delta)$ represents the discounted value at time $h$ of the shares that player $i$ receives at time $h$ and on. In particular, $x_{*}(i,\delta)=x^{1}_{*}(i,\delta)$. Similarly, define the game $v^{h}_{*}(\cdot,\delta)$ as

where the maximum is taken over all feasible sequences $x^{N}_{1},x^{N}_{2},\dots x^{N}_{h},z^{S}_{h+1},z^{S}_{h+2}\dots$ that coincide with $x^{N}_{1},x^{N}_{2},\dots$ up to $h$, the time when $S$ decides to leave.

A sequence of allocations $x^{N}_{1},x^{N}_{2},\dots$ is in the stable core if at any time $h$ the value of the shares of coalition $S$ exceeds the value of what coalition $S$ could guarantee in autarky, meaning without being engaged with others.

The following result applies to games where $v(N;x^{N})(N)$ is equal to $x^{N}(N)$ for every allocation $x^{N}$.

For the next definition we recall that the set of characteristic functions is a vector space. Denote by $\Delta(d)$ the set of stage allocations of games where the worth of the grand coalition is $d$. That is, $\Delta(d)=\{x^{N};x^{N}(N)=d,x^{N}(i)\geq B$ for every $i\in N\}$. The set $\Delta(d)$ is obviously compact, which is the reason why we impose the constraint that $x^{N}(i)\geq B$ for every $i\in N$.

Let $x\in\Delta(d)$ be an allocation. We say that $(y_{1},\dots,y_{k};\alpha_{1},\dots,\alpha_{k})$ is a split of $x$, if

where $\alpha_{j}\geq 0$, $\sum_{j=1}^{k}\alpha_{j}=1$, and $y_{j}$ is an allocation $j=1,\dots,k$. That is, $x$ is a convex combination of the allocations $y_{j}\in\Delta(d)$ with $\alpha_{j}$ being the respective weights.

If $V(N;\cdot)$ is continuous, then $\operatorname{conv}V(N;x)$ is closed and therefore, $\operatorname{conv}V(N;x)$ also contains all games of the form $\sum_{j=1}^{\infty}\alpha_{j}V(N;y_{j})$, where $x$ is expressed as an infinite convex combination of allocations: $x=\sum_{j=1}^{\infty}\alpha_{j}y_{j}.$

Before we get to the proof we wish to comment on the contents of this theorem. Just like in the folk theorem of the non-cooperative game theory, it characterizes the solution of the dynamic game in static terms. Specifically, it characterizes when the $\varepsilon$-fair core of the dynamic game is not empty in terms of the convexification of $V(N;\cdot)$.

Note that we refer to the $\varepsilon$-fair core rather than to the fair core. The question is whether we do it because we just cannot prove anything stronger, or that it is due to a structural insurmountable difficulty. Recall that the dynamic game is described by the dynamics, $V$ and by an initial game played at the first stage. While there is some control of future games through past allocations, there is no way to alter the initial game. It might happen that this stage-game hinders the existence of an exact core while $\varepsilon$-fair core does exist.

Let $M$ be a closed set of allocations. For a vector $x$ define $\operatorname{conv}_{M}V(N;x)$ as $\operatorname{conv}V^{N}(x)$ was defined, with the extra condition that the allocations $y_{j}$ are in $\operatorname{conv}M$. That is, $\operatorname{conv}_{M}V(N;x)$ is the set all games that can be expressed as $\sum_{j=1}^{k}\alpha_{j}V(N;y_{j})$, where $(y_{1},\dots,y_{k};\alpha_{1},\dots,\alpha_{k})$ is a split of $x$ and $y_{j}\in\operatorname{conv}M$, $j=1,\dots,k$.

The proof is similar to the proof of Theorem 5.2 and is therefore omitted.

Recall that $V(S;x^{S})$ is the stage game played with $S$ as the grand coalition after a stage in which the allocation was $x^{S}$. We now assume that $V(S;x)(T)$ depends on $x^{S}(T)=\sum_{i\in T}x_{i}$ for every $S$ that contains $T$ in a continuous and monotonically increasing fashion. In particular, the worth of coalition $T\subseteq S$ at time $t$ depends only on its total share at time $t-1$. For every coalition $T$ and time $t$, we define $U^{t}_{T}(c)$, inductively. $U^{1}_{T}(c)=V(S;x)(T)$, where $x(T)=c$. Note that this is well defined, as $V(S;x)(T)$ depends solely on $x(T)$. Then, $U^{t}_{T}(c)=U^{1}_{T}(U^{t-1}_{T}(c))$. Define $f_{T}(c)$ to be the limit of $U^{t}_{T}(c)$. Due to continuity this limit exists. Thus, it satisfies $f_{T}(c)=f_{T}(f_{T}(c))$. That is, $f_{T}(c)$ is a fixed point of $U^{1}_{T}$ and of $f_{T}$.

We further assume that $f_{T}(c)$ is finite for every $T$ and $c$, which in equivalent to assuming that either the set of fixed points of $U^{1}_{T}$ is unbounded or $U^{1}_{T}(x)<x$ for $x$ sufficiently large.²²2The function $U^{1}_{T}$ is continuous and monotonic. In case the set of the fixed points of $U^{1}_{T}$ is unbounded, every non-fixed point of $U^{1}_{T}$ is between two fixed points. The set of fixed points of $U^{1}_{T}$ is closed, and therefore for every non-fixed point of $U^{1}_{T}$, say $c$, there are two closest fixed point, one above $c$ and one below it. The sequence $U^{t}_{T}(c)$ then converges to one of the two (depending on whether $U^{t}_{T}(c)>c$ or $U^{t}_{T}(c)<c$). If, however, the set of fixed points of $U^{1}_{T}$ is bounded, then the sequence $U^{t}_{T}(c)$ diverges to infinity in case $U^{t}_{T}(c)>c$ asymptotically. In case $U^{t}_{T}(c)<c$ asymptotically, the sequence $U^{t}_{T}(c)$ is decreasing and $f_{T}(c)$ is finite.

Let $x$ be an allocation of $v_{1}$ and define the characteristic function $u_{x}$ as follows: $u_{x}(T)=f_{T}(x(T))$ for every coalition $T$.