Heterogeneously Perceived Incentives in Dynamic Environments:
Rationalization, Robustness and Unique Selections^††thanks: August 12, 2025. Contact: evan.piermont@rhul.ac.uk (Piermont), p.zuazogarin@hse.ru (Zuazo-Garin)

An extensive form consists in a list $\Gamma=\left\langle I,(A_{i})_{i\in I},H,Z\right\rangle$ where $I$ is s finite set of players, and:

• •

  For each player $i$, $A_{i}$ is a finite set of actions. A history represents the unfolding of the game and consists in a finite sequence of possibly simultaneous choices, i.e., on a finite sequence of elements from $\{h^{0}\}\cup A$, where $A:=\bigcup_{J\subseteq I}A_{J}$ and $A_{J}:=\prod_{i\in J}A_{i}$ for any $J\subseteq I$. We say that history $h^{\prime}$ follows history $h$, denoted by $h\preceq h^{\prime}$, if $h^{\prime}$ obtains from adding finitely many possibly simultaneous choices to $h$.¹⁰¹⁰10That is, when there exists some $(a^{n})_{n\leq N}\subseteq A$ such that $h^{\prime}=(h;(a^{n})_{n\leq N})$.

• •

  $H$ and $Z$ are finite and disjoint sets of histories such that $(H\cup Z,\preceq)$ is a rooted and oriented tree with terminal nodes $Z$. Symbol $h^{0}$ denotes the ex ante stage of the game, i.e., the root of the tree, and histories in $H$ and $Z$ are referred to as partial and terminal, respectively. For any player $i$ and partial history $h$, let $A_{i}\left(h\right)$ denote the set of actions available to $i$ at $h$. Player $i$ is active at $h$ if $A_{i}(h)$ is nonempty; let $H_{i}$ denote the set of these histories. We assume that: $(i)$ a player is never the only active one twice in a row, and $(ii)$ whenever a player is active, at least two actions are available to her.

In this context, the set of player $i$’s strategies is $S_{i}:=\prod_{h\in H_{i}}A_{i}(h)$ and, as usual, the set of strategy profiles is denoted by $S:=\prod_{i\in I}S_{i}$ and the set of player $i$’s opponents strategies, by $S_{-i}:=\prod_{j\neq i}S_{j}$. Obviously, for each partial history $h$ each strategy $s$ induces a unique terminal history, $z(s|h)$. Finally, let $S_{i}(h)$ and $S_{-i}(h)$ denote, respectively, the set of player $i$’s strategies and the set of $i$’s opponents’ strategies that reach partial history $h$, and $H_{i}(s_{i})$, the set of player $i$’s histories that can be reached when she chooses strategy $s_{i}$.¹¹¹¹11To be precise, $S_{i}(h)=\{s_{i}\in S_{i}|h\preceq z(s_{-i};s_{i}|h^{0})\textup{ for some }s_{-i}\in S_{-i}\}$ and $S_{-i}(h)=\prod_{j\neq i}S_{j}(h)$ on the one hand, and $H_{i}(s_{i}):=\{h\in H_{i}|s_{i}\in S_{i}(h)\}$ on the other.

A standard payoff structure (for extensive form $\Gamma$) consists of a list $\Upsilon:=(\Theta_{0},(\Theta_{i},u_{i})_{i\in I})$, where:

• •

  $\Theta_{0}$ is a compact and metrizable set of states of nature and, for each player $i$, $\Theta_{i}$ is a compact and metrizable set of payoff types. We denote the set of payoff states by $\Theta:=\Theta_{0}\times\prod_{i\in I}\Theta_{i}$ and, for each player $i$, the set of profiles of player $i$’s opponents’ payoff types by $\Theta_{-i}:=\prod_{j\neq i}\Theta_{j}$.

• •

  For each player $i$, $u_{i}:Z\times\Theta\rightarrow\mathds{R}$ is player $i$’s continuous utility function.¹²¹²12Throughout the paper finite sets are endowed with the discrete topology, $\mathds{R}$ is endowed with the Euclidean topology, product sets are endowed with the product topology and spaces consisting of closed sets, with the Hausdorff metric. For a given topological space $X$, its corresponding set of measures over the Borel $\sigma$-algebra is endowed with the weak$\ast$ topology. Unless stated otherwise, measurability always refers to Borel measurability.

The set of all standard payoff structures (for extensive form $\Gamma$) is denoted by $\mathscr{P}$.¹³¹³13A more general formalization of standard payoff structures allows for the set of payoff states not being a Cartesian product. We note that the whole theory developed in the paper extends to that case by introducing the necessary obvious changes. Thus, within a dynamic game with incomplete information $\langle\Gamma,\Upsilon\rangle$, each payoff state $\theta\in\Theta$ fully characterizes the profile of players’ utility functions: $(u_{i}(\,\cdot\,,\theta))_{i\in I}$. Besides, each player $i$ is assumed to know her payoff type $\theta_{i}$ and (possibly) face uncertainty about the rest of components of $\theta$. Hence, besides the description of preferences, each payoff state $\theta$ also determines the information players have about the payoff states: In every moment of the game, a player $i$ with payoff type $\theta_{i}$ knows that the true payoff state is some element in $\{\theta_{i}\}\times\Theta_{0}\times\Theta_{-i}$; in consequence, $\{\theta_{i}\}\times\Theta_{0}\times\Theta_{-i}$ is, precisely, the set of payoff states that player $i$ can eventually deem as possible as the game unfolds.

The questions we explore throughout the paper are related to the strategic impact of small relaxation of players’ common knowledge of an information structure $\Upsilon$. We develop this point in detail in Section 3.1, but before introducing it we need to first clarify what is understood as a small, or perturbation, in a standard payoff structure. The fact that the sets of payoff states can vary vastly across different standard payoff structures and be of radically different nature presents some inconvenience for an analyst interested in assessing how close two different standard payoff structures are form each other. To circumvent this issue we introduce a canonical representation that allows for envisioning every standard payoff structure as part of an object directly derived from the extensive form:

The interpretation of each element $\upsilon=(\upsilon_{0},(\upsilon_{i})_{i\in I})\in\mathcal{C}(\Upsilon)$ is similar to that of each payoff state $\theta$: A description of each player $i$’s preferences and information. First, $\upsilon_{0}$ consists in a profile specifying a utility function $(\upsilon_{0})_{i}:Z\rightarrow\mathds{R}$. Second, $\upsilon_{i}$ can be interpreted as the set of profiles that can be conceived by a player $i$ already endowed with a payoff type as the game unfolds—anything outside of $\upsilon_{i}$ will never be part of $i$’s beliefs (even if necessary for rationalizing observed behavior).

Now, the main reason for introducing Definition 1 is that it allows for envisioning payoff structures as informational constraints over a canonical object directly derived from the extensive form, namely, the space of profiles of all conceivable utility functions, $\prod_{i\in I}\mathds{R}^{Z}$: Simply notice that, given $\Upsilon$, $\mathcal{C}_{0}(\Upsilon)$ is a compact subset of $\prod_{i\in I}\mathds{R}^{Z}$ and that for each $i\in I$, each $\upsilon_{i}\in\mathcal{C}_{i}(\Upsilon)$ is a compact subset of $\mathcal{C}_{0}(\Upsilon)$—so that $\mathcal{C}_{i}(\Upsilon)$ is a compact subset of compact subsets of $\mathcal{C}_{0}(\Upsilon)$. The notion of convergence for canonical representations of standard payoff is very natural: Say that $(\mathcal{C}(\Upsilon^{n}))_{n\in\mathds{N}}$ converges to $\mathcal{C}(\Upsilon)$ if:

• 1.

  Sequence $(\mathcal{C}_{0}(\Upsilon^{n}))_{n\in\mathds{N}}$ converges to $\mathcal{C}_{0}(\Upsilon)$ in the Hausdorff metric for compact subsets of $\prod_{i\in I}\mathds{R}^{Z}$.

• 2.

  For each player $i$, sequence $(\mathcal{C}_{i}(\Upsilon^{n}))_{n\in\mathds{N}}$ converges to $\mathcal{C}_{i}(\Upsilon)$ in the Hausdorff metric for compact subsets of compact subsets of $\prod_{i\in I}\mathds{R}^{Z}$

The use of canonical representations greatly simplifies (at a conceptual level) our analysis of robustness in Section 4; perturbing payoff structures will consist on simply perturbing profiles of sets instead of perturbing both sets and utility functions whose domain includes these perturbed sets:

Next, we introduce the main new formal element of the paper. A subjective payoff structure is a hierarchical construction that represents how a player perceives the payoff structure of a game, how she perceives her opponents to perceive the payoff structure of the game, and so on. Thus, technically, a subjective payoff structure consists in a list in which the first component is a standard payoff structure (the one the player considers to be the right one), the second component is a map that assigns a subjective payoff structure to each opponent (the one the player considers each opponent to consider the right one), and so on. One consistency condition is required (see part $(iii)$ in the definition below): If player $i$ considers that $\Theta_{j}$ are the possible payoff types of player $j$, then she must consider that $j$ considers that $\Theta_{j}$ is (at least in terms of canonical representations) the set of her own possible payoff types—i.e., there is common knowledge in the event that players know their own set of payoff types.

Throughout the paper, with some abuse of notation, we will indistinctly use $\Upsilon$ to refer to a standard payoff structure or to the subjective payoff structure that represents common knowledge of $\Upsilon$; accordingly, we will treat $\mathscr{P}$ as a subset of $\mathscr{P}_{i}^{\infty}$.¹⁵¹⁵15That is, for player $i$, $d_{i}^{\Upsilon}$ such that $d_{i,1}^{\Upsilon}=\Theta$, $d_{i,2}^{\Upsilon}(j)=d_{j,1}^{\Upsilon}$, $d_{i,3}^{\Upsilon}(j)=d_{j,2}^{\Upsilon}$, and so on. For a given subjective payoff structure $d_{i}$ we denote:

• •

  By $d_{j|i}:=(d_{i,k}(j))_{k\geq 2}$ and $d_{-i|i}:=(d_{j|i})_{j\neq i}$, the subjective payoff structure ascribed by $d_{i}$ to player $j\neq i$, and the list of subjective payoff structures ascribed by $d_{i}$ to every player $j\neq i$, respectively.

• •

  By $(d_{i,1})_{0}$, the set of states of nature that corresponds to the set of payoff states in standard payoff structure $d_{i,1}$, and by $(d_{i,1})_{j}$, for each $j\in I$, the set of player $j$’s payoff types in that same set.

Since subjective payoff structures put restrictions on which payoff states can be considered by the player, obviously, not every type $t_{i}$ is necessarily consistent with every subjective payoff structure $d_{i}$; in order for the pair $(d_{i},t_{i})$ be consistent, the payoff type of $t_{i}$ must be in $(d_{i,1})_{i}$, the support of its first-order belief must be included in the projection on $(d_{i,1})_{0}\times(d_{i,1})_{-i}$ of $d_{i,1}$, and so on.¹⁶¹⁶16Formally, we require that $(1)$ $(d_{i,1})_{i}$ contains $\theta_{i}(t_{i})$; $(2)$ that $\textup{Proj}_{(d_{i,1})_{0}\times(d_{i,1})_{-i}}(d_{i,1})$ contains the support of $\pi_{i,1}(t_{i})$; $(3)$ that for every $t_{-i}$ in the support of $\tau_{i}(\pi_{i}(t_{i}))$ and every $j\neq i$, $(1)$ and $(2)$ hold w.r.t. $d_{j|i}$; and so on. For each profile of subjective payoff structures $d$, we let $T_{i}(d_{i})$ and $T_{-i}(d_{-i}):=\prod_{j\neq i}T_{j}(d_{j})$ denote, respectively, the set of player $i$’s types that are consistent with $d_{i}$, and the set of opponents types that are consistent with $d_{-i}$. Finally, let $T(d):=T_{i}(d_{i})\times T_{-i}(d_{-i})$. Consistent combinations of subjective payoff structures and types play a central role in our analysis, since they fully specify players’ perception of the incentives of the game:

Thus, a subjective model fully specifies the player’s perception of the payoff structure of the game; the subjective payoff structure describes the information about preferences the player will restrict her attention to, and the type, her initial probabilistic assessment about the payoff states. If the player perceives the set of states as commonly known, her perception is represented by a standard subjective model and, if every player agree on their perception of the set of payoff states and this fact is commonly known, players’ perceptions are represented by a standard model. We present now an example aimed at illustrating these notions.

Consider the following extensive form:

The mechanics of the game are simple: There are two players ($1$ and $2$) with two available actions each (to advance and to go down); if, at her turn, a player chooses to go down the game ends, if, instead, she chooses to advance it is then the other player’s turn. Actions are observed, player $1$ chooses first and has at most two turns, and player $2$ has at most only one turn. Now, by adding a standard payoff structure $\Theta^{(n,+)}$, we can have a standard dynamic game as the following one, in which for a given $n$, utility functions are commonly known:

Now, if we want to introduce the possibility of discrepancies among players, we can assume that player $2$’s subjective payoff structure is $d_{2}^{n}$, representing common knowledge of $\Theta^{(n,+)}$,¹⁷¹⁷17I.e., such that $d_{2,1}^{n}=\Theta^{(n,+)}$, $d_{2,2}^{n}(1)=\Theta^{(n,+)}$, $d_{2,3}^{n}(1)(2)=\Theta^{(n,+)}$, etc. and that player $1$’s subjective structure is given by $d_{1}^{n}$, with a first component representing the following payoff structure, $\Theta^{(n,-)}$:

and such that $d_{1}^{n}$ considers that player $2$ considers $\Theta^{(n,+)}$ to be commonly known.¹⁸¹⁸18I.e., such that $d_{1,1}^{n}=\Theta^{(n,-)}$ and $d_{2|1}^{n}=d_{2}^{n}$. Thus, $(d_{1},d_{2})$ represent a situation in which player $2$ considers that $\Theta^{(n,+)}$ is commonly known and player $1$ knows this, but considers the payoff structure to be $\Theta^{(n,-)}$. For each subjective payoff structure, there is a unique type for the player consistent with it: For player $2$, the one that represents common belief in the payoffs given by the unique state in $\Theta^{(n,+)}$, $t_{2}^{n}$, and for player $1$, the one that represents probability 1 belief in the payoffs given by the unique state in $\Theta^{(n,-)}$ and in type $t_{2}^{n}$, $t_{1}^{n}$. Thus, $M_{1}^{n}:=(d_{1}^{n},t_{1}^{n})$ and $M_{2}^{n}:=(d_{2}^{n},t_{2}^{n})$ are the unique possible subjective models; the first one is not standard, the second one is. Thus, $(M_{1}^{n},M_{2}^{n})$ is not a standard model.

Richness assumptions play a central role in the literature of structure theorems and unique selections (see Weinstein and Yildiz, 2007, Penta, 2012 or Chen, 2012). The reason is that states in which a strategy is dominant are employed as ‘seeds’ to initiate a contagion argument that propagates through the hierarchy of beliefs: A player might not believe in one of those states, but if she believes that her opponent believes in one of then, then the player should choose a best response to the strategy her opponent considers to be dominant. Formally, richness is defined as follows:

Notice that in the case of standard payoff structures, the notion is rather demanding. It requires that it is common knowledge among players that no strategy is known not to be conditionally dominant. Or, in other words, that it is commonly known among players that every strategy can be found to be conditionally dominant for some appropriate belief about the payoff states. On the contrary, subjective payoff structures allow for greatly relaxing this common knowledge assumption while keeping its bite. Players may know that richness does not hold, but they may consider that their opponents consider it to hold; or they may consider that it does not hold, that their opponents consider it not to hold, but consider that their opponents consider their opponents to consider that it holds. Thus, it is possible to approximate common knowledge of absence of richness while still having the chance to exploit an infection argument: I could perceive the game as not being rich, but perceive that my opponents perceive it to be rich, or only perceive that they perceive that their opponents perceive that it is rich, and so on. In order to formalize the idea that richness holds at high orders but not low ones let us introduce the following definition:

Notably, it turns out that higher-order richness is a generic property within the space of subjective payoff structures; thus, while richness is a very demanding notion for games with a commonly known state space, higher-order richness is extremely mild for games with subjective payoff structures:

We would like to be emphatic about how natural this separation of rules and stakes is: While the possible outcomes of an economic interaction can be objectively described, there is nothing intrinsically objective in the notion of utility; the allocation of resources that results from each combination of reported preferences can be commonly known, the market shares that correspond to each different combination of pricing strategies can be commonly known, etc. But each individual agent’s preference over material outcomes is a personal trait of hers. In consequence, it seems reasonable to remove the set of states, or possible utilities, from the objective (in the sense commonly known) part of the game

Subjective payoff structures allow for relaxing all common knowledge assumptions about payoffs in a simple, tractable way; the only common knowledge assumptions they entail is the trivial one: The set of utility functions players reason about consists in functions that map outcomes to utilities. Furthermore, not only do subjective payoff structures allow for approximating models not satisfying any richness assumption with models that satisfy richness at high orders, but they also allow for approximating models satisfying certain knowledge assumptions (say, the no information case, as in Weinstein and Yildiz, 2007, or Chen, 2012) with models that satisfy different knowledge assumptions at high orders (say, models with persistent private information, as in Penta, 2012). In this sense, subjective payoff structures allow for enhancing the subjective nature of individual strategic decisions and, following the ‘Wilson doctrine’, develop game-theoretic analyses that dispense with unrealistic, excessively demanding common knowledge assumptions.

To the best of our knowledge, the methods for reducing common knowledge assumptions closest to our approach build on Battigalli and Siniscalchi’s (2003) $\Delta$-restrictions. Formally, a $\Delta$-restrictions is a specification of profiles of first-order beliefs, $\Delta:=(\Delta_{i})_{i\in I}$, so that for each player $i$ we have a collection $\Delta_{i}:=(\Delta_{\theta_{i}})_{\theta_{i}\in\Theta_{i}}$ where $\Delta_{\theta_{i}}\subseteq\Delta^{H_{i}\cup\{h^{0}\}}(S_{-i}\times\Theta_{0}\times\Theta_{-i})$ for every payoff type $\theta_{i}$. The difference between $\Delta$-restrictions and profiles of subjective payoff structures is mainly twofold. On the one hand, $\Delta$-restrictions allow for placing restrictions on the evolution of strategic uncertainty, not only payoff uncertainty. On the other, $\Delta$-restrictions impose constraints over higher-order beliefs by assuming that $\Delta$ is commonly known among players. Thus, it is possible to envision subjective payoff structures as $\Delta$-restrictions that $(1)$ do not place constrains on beliefs about opponents’ behavior, and $(2)$ include specifications of constraints on higher-order beliefs that do not necessarily entail nontrivial common knowledge assumptions.

The restrictions embodied by subjective payoff structures also provide a reduced from method of representing unawareness in games. The uncertainty about the payoff state could be thought of as arising from the resolution of some unmodeled contingencies regarding physical aspects of the world external to the choices made by the players—a state of nature $\theta_{0}\in\prod_{i\in I}\mathds{R}^{Z}$ being shorthand for the particular resolution of this uncertainty that induces such payoffs to each player. Thus, although a player may easily envision the entirety of the abstract space $\prod_{i\in I}\mathds{R}^{Z}$, in actuality he considers on those states relating to some contingency he is aware of.

Now, when a player is aware of a contingency but initially deems it probability zero, he may revise his beliefs by reasoning about his opponents behavior. This is not so when the player is unaware of the contingency, since observing his opponents actions will not (in general) enlighten him to new contingencies. Thus, the essential attribute of our formalism—persistent, heterogeneous belief about the set of payoff states—can arise as a natural consequence of asymmetric awareness.

Of course, an agent’s perception of other agents’ awareness must be expressible in terms of things he is himself aware of. This is the motivation of various extant restrictions in the awareness literature—closure under subformula in Fagin and Halpern (1988), the confinidness property in Heifetz, Meier and Schipper (2006), property F1 in Piermont (2019). The translation to our set up would be: if $i$ is aware that $j$ is aware of $\theta$ then $i$ must be himself aware of $\theta$. This conceptual restriction can be made formal via the edict: $(d_{i,2}(j))_{0}\subseteq(d_{i,1})_{0}$. The set of payoff states that $i$ considers $j$ to consider is a subset of those he himself is aware of.

The evolution throughout the game of each player’s subjective beliefs is presented by conjectures. Formally, for each player $i$ endowed with a subjective model $(d_{i},t_{i})$, a conjecture consists in a list of beliefs $\mu_{i}=(\mu_{i}(h))_{h\in H_{i}\cup\{h^{0}\}}$ satisfying that:

• $(i)$

  For every history $h$, either initial or in which player $i$ is active, $\mu_{i}(h)$ is a probability measure on $S_{-i}\times(d_{i,1})_{0}\times T_{-i}(d_{-i|i})$ that assigns probability $1$ to $S_{-i}\times(d_{i,1})_{0}\times T_{-i}(d_{-i|i})$.

• $(ii)$

  Whenever possible, beliefs are updated following conditional probability.

• $(iii)$

  The initial beliefs represented by $\mu_{i}$ are consisted with those represented by $t_{i}$:

  $$\textup{marg}_{(d_{i,1})_{0}\times T_{-i}(d_{-i|i})}\mu_{i}(h^{0})=\tau_{i}(\pi_{i}(t_{i})).$$

We denote the set of conjectures consistent with subjective model $(d_{i},t_{i})$ by $\textup{C}(d_{i},t_{i})$.

Given a payoff type $\theta_{i}$ and a history $h\in H_{i}\cup\{h^{0}\}$, each conjecture $\mu_{i}$ naturally induces a conditional expected payoff for each strategy $s_{i}$:

$$U_{i}\left(\mu_{i},s_{i}\left|\theta_{i},h\right.\right):=\int_{S_{-i}\times\Theta_{0}\times\Theta_{-i}}u_{i}\left(z\left(s_{-i};s_{i}\left|h\right.\right),(\theta_{0},\theta_{i},\theta_{-i})\right)\textup{d}(\textup{marg}_{S_{-i}\times\Theta_{0}\times\Theta_{-i}}\mu_{i}(h)).$$

Finally, given a payoff type $\theta_{i}$ and a conjecture $\mu_{i}$, the notion of sequential rationality is then captured via the set of (sequential) best responses:¹⁹¹⁹19Notice that, if we denote the set of all conjecture of player $i$ as $\Delta^{H_{i}\cup\{h^{0}\}}(S_{-i}\times(d_{i,1})_{0}\times T_{-i}(d_{-i|i}))$, then $r_{i}:\Theta_{i}\times\Delta^{H_{i}\cup\{h^{0}\}}(S_{-i}\times(d_{i,1})_{0}\times T_{-i}(d_{-i|i}))\rightrightarrows S_{i}$ is upper hemicontinuous and nonempty-valued.

$$r_{i}\left(\theta_{i},\mu_{i}\right):=\left\{s_{i}\in S_{i}\left|s_{i}\in\bigcap_{h\in H_{i}(s_{i})}\textup{arg}\underset{s^{\prime}_{i}\in S_{i}}{\textup{max}}\,U_{i}\left(\mu_{i},s^{\prime}_{i}\left|\theta_{i},h\right.\right)\right.\right\}.$$

Extensive form rationalizability formalizes the notion of forward induction; that is, the idea that while forming conjectures about opponents’ behavior, players take observed past choices into account. Specifically, as clarified by the epistemic analysis by Battigalli and Siniscalchi (2002), players rationalize observed behavior to the ‘highest possible degree’ of strategic sophistication the behavior is consistent with: Rationality; or rationality and belief whenever possible in opponents’ rationality; or rationality, belief whenever possible in opponents’ rationality and belief whenever possible in opponents’ belief whenever possible in their opponents’ rationality; etc. Thus, the main conceptual feature of the solution concept is that observed behavior is employed to infer other players’ private information and that conjectures about behavior are based on this inference. To be more specific, if a player entertains certain beliefs about the payoff type of an opponent, upon observing a choice that would have been irrational for that payoff type, if possible, she should update her beliefs about the payoff type and believe that the true type is one for which the observed behavior is rational.

Formally, the original notion due to Pearce (1984) and Battigalli (1997) is generalized, in our context, as a correspondence that maps each player $i$’s subjective model $(d_{i},t_{i})$ to a certain subset of strategies $\textup{F}_{i}(d_{i},t_{i})\subseteq S_{i}$:

Backward rationalizability (see Penta, 2011, 2015) extends the standard notion of backward induction to games with incomplete information. The reasoning process behind differs from that behind extensive form rationalizability seen above: Here, upon observing unexpected behavior, players do not necessarily try to find a rationale for said behavior (even at the cost of dropping their belief in ‘common belief in rationality’); instead, they can interpret observed behavior as a mistake and maintain their belief that ‘common belief in rationality’ holds.²⁰²⁰20It is not conceptually accurate to talk about common belief in rationality in a dynamic game, not at every history this belief can be held; however, we keep this terminology in the present discussion for expositional simplicity. Thus, the difference between the extensive form rationalizability and backward rationalizability is one of epistemic priority (see Catonini, 2019): While the former dictates to update higher-order beliefs about payoff states in order to perform a rationalization to the highest possible degree, the latter permits to maintain the higher-order beliefs about payoff states, believe that unexpected behavior is uninformative and keep the faith on continuation behavior being consistent with rationality and common belief thereof.

Under subjective payoff structures, Penta’s (2011; 2015) original notion is generalized as a correspondence that maps each player $i$’s subjective model $(d_{i},t_{i})$ to a certain subset of strategies $\textup{B}_{i}(d_{i},t_{i})\subseteq S_{i}$. It is easy to check that the following Definition 8 below captures the idea, formalized in epistemic language by Perea (2014) and Battigalli and De Vito (2018),²¹²¹21Unlike Penta (2011, 2015), and in order to make the comparison with extensive form rationalizability more immediate, we drop the requirement that a player has beliefs about her own strategy. Nonetheless, it is immediate that the version we employ captures the idea of ‘continuation behavior consistent with common belief in rationality’ as formalized by rationality and common belief in future rationality in Perea (2014) or common full belief in optimal planning and in belief in continuation consistency in Battigalli and De Vito (2018). that players believe that rationality will hold at every possible higher-order in the continuation game, even after moves inconsistent with it. First, define the set of profiles of player $i$’s opponents’ strategies that are equivalent to $s_{-i}$ in the continuation game following history $h$ as:

$$[s_{-i}]_{h}:=\left\{s_{-i}^{\prime}\in S_{-i}\left|\forall j\neq i,\forall h\in H_{j}\textup{ s.t. }h\prec h^{\prime},s^{\prime}_{j}(h)=s_{j}(h)\right.\right\}.$$

Then, backward rationalizability is defined in our context as:

Consider again the extensive form studied in Section 3.1.2. Let us endow it with a standard payoff structure $\Theta=\{\theta\}$ such that the resulting dynamic game is:

Now, for each player $i=1,2$ let $d_{i}^{0}$ represent the standard subjective payoff structure associated to $\Theta$. Clearly, there is a unique type for player $i$ consistent with $\Theta$, namely the one representing common belief in $\theta$, which we denote by $t_{i}^{0}$. Thus, there is a unique profile of subjective models, $(M_{1},M_{2})$ where $M_{i}=(d_{i}^{0},t_{i}^{0})$ for both $i=1,2$, which is, indeed, a standard model. The strategic analysis of this game yields the following conclusions:

• •

  The unique prediction consistent with extensive form rationalizability is $D_{1}$ (outcomewise). This is easy to see. First, player $1$ will never choose $A_{2}$ at her second turn. Second, if player $2$ finds herself at $A_{1}$, while she can be surprised that $1$ took the risk of advancing instead of guaranteeing a payoff of $2$, she need not abandon the belief that 1 is rational. Player $1$ may have advanced, rationally, by believing that $2$ does not expect her (i.e., 2 does not expect 1) to be rational in her second turn, and therefore believing that $2$ will advance as well. Thus, since $2$ maintains the belief that $1$ is rational, she will expect $1$ to choose $D_{2}$ in her first second turn, and will therefore choose $d$. Being able to forecast this, player $1$ will choose $D_{1}$ in her first turn.

• •

  The unique prediction consistent with backward rationalizability is $D_{1}$ as well. This is immediate.

Thus, both solution concepts uniquely predict $D_{1}$ is played. Suppose now that we perturb the game using a sequence consisting of the subjective models defined in Section 3.1.2, $(M_{1}^{n},M_{2}^{n})_{n\in\mathds{N}}$. For each $n\in\mathds{N}$, the situation can be represented as follows:

Remember that here players commonly agree on the payoffs that correspond to terminal histories $(A_{1},d)$, $(A_{1},a,D_{2})$ and $(A_{1},d,A_{2})$. However, in player $2$’s mind they also commonly agree about the payoffs corresponding to $D_{1}$ being $(2+1/n,0)$, but this is wrong: In player $1$’s mind, the payoff that corresponds to $D_{1}$ is $(2-1/n,0)$, but she knows that player $2$ believes in the commonly agreement on $(2+1/n,0)$. Obviously, the larger $n$, the closer we are to the game with commonly known payoff structure $\Theta$. This is the sense in which we say that $(M_{1}^{n},M_{2}^{n})_{n\in\mathds{N}}$ converges to $(M_{1},M_{2})$. For each given $n$, the corresponding strategic analysis yields the following conclusions:

• •

  The predictions consistent with extensive form rationalizability are multiple now: $D_{1}$, $(A_{1},d)$ and $(A_{1},a,D_{2})$. This is easy to see. Upon observing $A_{1}$, which in her mind is strictly dominated, player $2$ may drop the belief that player $1$ is rational, and may have any arbitrary belief about 1’s future behavior. She might believe that player $1$ will be completely irrational and choose $A_{1}$ in her second turn, or she might believe player $1$ will be more sensible next time and choose $D_{2}$. In consequence, both $a$ and $d$ are rationalizable choices for player $2$. As a result, $(A_{1},D_{2})$, $(D_{1},A_{2})$ and $(D_{1},D_{2})$ are all rationalizable strategies for player $1$, depending on what she expects player $2$ to choose.

• •

  The unique prediction consistent with backward rationalizability is $D_{1}$ again. Also again, this is immediate.

The example shows that extensive form rationalizability refining backward rationalizability crucially depends on players commonly agreeing on the set of possible payoff states. As arbitrarily small disagreements arise, both $(A_{1},d)$ and $(A_{1},a,D_{2})$ become consistent with extensive form rationalizability; however, none of them is consistent with backward rationalizability. Importantly, the example delivers an additional lesson: Extensive form rationalizable outcomes fail to be upper hemicontinuous, even in standard models; all $D_{1}$, $(A_{1},d)$ and $(A_{1},a,D_{2})$ are consistent with extensive form rationalizability along the sequence, but only $D_{1}$ is so in the limit.