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Perfect Bayesian equilibrium

In game theory, a perfect Bayesian equilibrium (PBE) is a refinement of Bayesian Nash equilibrium for extensive-form games with incomplete information, where players' strategies and beliefs must satisfy sequential rationality at every information set and beliefs are derived via Bayesian updating wherever possible.^[1] Formally, a PBE consists of a complete plan of action (behavior strategy) for each player and a system of beliefs for each player at every information set, such that strategies maximize expected payoffs conditional on beliefs and others' strategies, while beliefs are consistent with the strategies using Bayes' rule on paths with positive probability.^[2] The concept was developed to address limitations in subgame-perfect Nash equilibrium when applied to games with imperfect or incomplete information, where "off-path" beliefs cannot always be uniquely determined by Bayes' rule due to zero-probability events.^[1] Introduced in the late 1980s by Drew Fudenberg and Jean Tirole, PBE imposes a "no-signaling-what-you-don't-know" condition on beliefs, ensuring that actions reveal only information that players actually possess, thus preventing implausible inferences in dynamic settings like signaling games.^[1] This makes PBE a weaker requirement than sequential equilibrium, which adds further consistency conditions on belief limits, but PBE remains equivalent to sequential equilibrium in simpler cases, such as games with at most two types per player or two periods of play.^[1] PBE is particularly useful in multi-stage games where actions are observed but types (private information) are not, such as entry deterrence or principal-agent models, as it ensures robustness against non-credible threats by requiring rationality throughout the game tree.^[2] Over time, refinements like those proposed by Joel Watson in 2015 have generalized PBE to broader classes of games, including infinite-horizon settings, by emphasizing "plain consistency" in belief updates across strategy dimensions without invoking limits of belief hierarchies.^[3] These developments highlight PBE's role as a foundational tool for analyzing strategic interactions under uncertainty, balancing tractability with behavioral realism.^[3]

Introduction

Definition and Motivation

Perfect Bayesian equilibrium (PBE) builds upon foundational concepts in game theory for modeling uncertainty and sequential decision-making. Bayesian games, introduced by Harsanyi in his seminal three-part series published between 1967 and 1968, capture situations of incomplete information where each player possesses a private type that influences payoffs, and players form beliefs about others' types according to a common prior.^[4] A Bayesian Nash equilibrium (BNE) in such games requires that each player's strategy maximizes their expected payoff, conditional on their type and beliefs about others' strategies and types. To extend this to dynamic settings, extensive-form games with perfect recall are employed; these represent sequential moves via a game tree where each player recalls all previous actions and information sets they have observed, ensuring that strategies can be behavioral (specifying actions at each decision point) rather than requiring full contingency plans over unreached histories. In these dynamic Bayesian games, a PBE is formally defined as a profile of behavioral strategies—one for each player—and a system of beliefs for each player over the types of others at every information set, satisfying two key conditions.^[5] First, sequential rationality holds: at every information set, whether reached in equilibrium or not, the prescribed action maximizes the player's expected payoff given their beliefs about types at that set and the continuation strategies of all players. Second, consistency requires that beliefs are updated using Bayes' rule along any equilibrium path where the information set has positive prior probability; off the equilibrium path, beliefs must be specified consistently with Bayes' rule to the extent possible, given the game's structure.^[5] The motivation for PBE arises from the limitations of BNE in sequential environments with asymmetric information. While BNE guarantees that no player can profitably deviate from their full strategy profile in expectation, it does not enforce optimality at individual information sets, particularly those not reached under the equilibrium strategies; this can sustain outcomes reliant on non-credible threats or responses to hypothetical deviations, as the normal-form representation aggregates payoffs without isolating subgames or unreached nodes.^[5] PBE refines BNE by incorporating sequential rationality everywhere, ensuring strategies remain incentive-compatible at every decision point given updated beliefs, which better captures realistic behavior in multi-stage interactions under uncertainty, such as signaling or bargaining with private information. A simple illustration of BNE's potential failure occurs in a basic static Bayesian game extended to sequential moves, such as a sender-receiver setup where the sender's private type leads to an off-equilibrium action (e.g., an unexpected signal). In BNE, the receiver's strategy might specify a suboptimal response to this signal—say, one that ignores the implications for the sender's type—because the overall expected payoff remains maximized along the equilibrium path, with arbitrary beliefs supporting the deviation's unprofitability. However, if the off-equilibrium signal occurs, the receiver's play becomes inefficient; PBE avoids this by mandating an optimal response at the corresponding information set, based on some consistent belief, preventing reliance on implausible off-path behavior.^[5]

Historical Context

The concept of Perfect Bayesian Equilibrium (PBE) emerged within the broader framework of Bayesian games, which were formalized by John C. Harsanyi in his seminal three-part series published between 1967 and 1968. Harsanyi's work addressed games of incomplete information by modeling players' uncertainty over opponents' types using Bayesian probability distributions, laying the groundwork for solution concepts that incorporate beliefs and updates via Bayes' rule. Building on this foundation, David M. Kreps and Robert B. Wilson introduced the related concept of sequential equilibrium in 1982, which requires strategies to be sequentially rational given consistent beliefs and serves as a refinement of Bayesian Nash equilibrium for extensive-form games with imperfect information.^[6] PBE, developed subsequently, offers a simpler alternative to sequential equilibrium by ensuring sequential rationality and belief consistency without the full limit-of-information-sets construction required for sequential equilibrium's stronger uniformity conditions. In 1987, In-Koo Cho and David M. Kreps explicitly named and developed PBE as a refinement tailored to signaling games, where it imposes consistency conditions on off-equilibrium beliefs to eliminate implausible equilibria.^[7] Subsequent refinements appeared in Drew Fudenberg and Jean Tirole's 1991 textbook Game Theory, which provided a formal definition of PBE for multi-stage games with observed actions and explored its relationship to sequential equilibrium, emphasizing practical consistency requirements for beliefs.^[5] Post-2000, extensions of PBE have integrated insights from behavioral game theory, incorporating bounded rationality and empirical deviations from standard assumptions, as discussed in Colin F. Camerer’s 2003 book Behavioral Game Theory: Experiments in Strategic Interaction.^[8] In the 2020s, PBE has seen applications in computational game theory for AI-driven environments, such as algorithms for approximating equilibria in sequential auctions.^[9]

Core Components

Strategies and Types

In perfect Bayesian equilibrium (PBE), strategies are formalized as behavioral strategies in dynamic games with incomplete information, where each player i specifies a probability distribution over actions contingent on their information set and private type. A behavioral strategy for player i, denoted \sigma_i, is a function that maps each information set I_i \in \mathcal{I}_i (where \mathcal{I}_i is the set of player i's information sets) to a probability distribution over the feasible actions A_i(I_i) at that set.^[3] This formulation arises from the perfect recall assumption, under which players remember all their past actions and observations, allowing strategies to depend only on the current information set rather than the full history of play.^[3] Types introduce private information into the game, representing each player's uncertain knowledge about the environment or opponents' characteristics, drawn from a finite type space \Theta_i for player i. Following Harsanyi's framework for Bayesian games, types \theta_i \in \Theta_i are realized according to a common prior probability distribution p(\theta) over the joint type space \Theta = \prod_{i \in N} \Theta_i, where N is the set of players, and players condition their actions on their own type while holding beliefs about others' types. In PBE, a full strategy profile must therefore be type-contingent, meaning \sigma_i: \mathcal{I}_i \times \Theta_i \to \Delta(A_i), where \Delta(A_i) denotes the set of probability distributions over actions, to account for how different types may lead to distinct behavioral responses.^[3] Payoffs in these settings are defined as expected utilities that incorporate both the strategy profile and the realized types, capturing the uncertainty inherent in incomplete information. Specifically, player i's payoff is given by u_i(\sigma, \theta) = \mathbb{E}[r_i(h, \theta) \mid \sigma, \theta], where r_i(h, \theta) is the von Neumann-Morgenstern utility from terminal history h under type profile \theta, and the expectation is taken over histories induced by the strategies \sigma = (\sigma_i, \sigma_{-i}).^[3] This structure ensures that types not only affect players' preferences but also serve as signals or sources of asymmetry, necessitating strategies that optimize expected payoffs across possible type realizations. The role of types in incomplete-information settings fundamentally alters strategic considerations, as they create uncertainty that players must resolve through type-dependent actions, preventing the direct observability of payoffs or opponents' incentives that characterizes complete-information games. In dynamic contexts, this requires strategies to be robust to the private information encoded in types, promoting equilibria where actions reveal or conceal type-related information over time without violating the perfect recall condition.^[3]

Beliefs and Information Sets

In the framework of Perfect Bayesian Equilibrium (PBE), the informational aspects of dynamic games with incomplete information are formalized through information sets and beliefs, which together describe the uncertainty faced by players at decision points. An information set for a player is defined as a collection of nodes in the extensive-form game tree at which that player moves, such that the player cannot distinguish between these nodes based on the information available to them. These sets must satisfy two conditions: the same player moves at every node in the set, and the available actions are identical across all nodes in the set. The representation assumes perfect recall, whereby players remember all their previous actions and the information they had when making those choices. This structure, introduced in the analysis of extensive games, ensures that strategies are contingent on the player's observable history up to the current decision point.^[3] Beliefs in PBE represent the subjective probability distributions that a player holds over the possible nodes or types consistent with their information set. At an information set I reached after a history h, the belief is denoted \mu(\cdot | h, I), which is a probability measure assigning probabilities to the elements of I (such as terminal nodes, opponent types, or hidden states). These beliefs encapsulate the player's updated assessment of the game's state given the observed history and their private information. In Bayesian settings, initial beliefs derive from a common prior probability distribution over the players' types, which is common knowledge among all players at the outset of the game. This common prior ensures that any differences in players' beliefs stem solely from asymmetric information about types, rather than divergent subjective probabilities.^[3] A key distinction in PBE concerns on-equilibrium and off-equilibrium beliefs. On the equilibrium path—i.e., at information sets reached when all players follow their equilibrium strategies—beliefs are uniquely determined by applying Bayes' rule to the common prior and the equilibrium strategies. In contrast, off-equilibrium beliefs, associated with unreached information sets, are not constrained by Bayes' rule due to the lack of observed data and can be specified arbitrarily, provided they are consistent with the overall equilibrium requirements. This flexibility allows for a range of possible PBE outcomes, as off-path beliefs influence players' incentives to deviate and must support optimal actions at those points.^[3]

Equilibrium Conditions

Sequential Rationality

Sequential rationality is a core requirement in perfect Bayesian equilibrium (PBE) that ensures players' strategies are optimal at every decision point in the game tree, conditional on the information available at that point. Specifically, for each player i and every information set I_i in their strategy space, the prescribed action must maximize the player's expected utility given their beliefs about the state of the world and the strategies of other players. This condition applies universally, including to information sets that are reached with zero probability under the equilibrium strategies, thereby enforcing rational behavior even in hypothetical or off-equilibrium scenarios.^[1] Formally, sequential rationality in a PBE demands that the strategy profile \sigma satisfies, for all players i, all information sets I_i, and all feasible actions a_i \in A(I_i),

\sigma_i(I_i) \in \arg\max_{a_i'} \sum_{\theta_{-i}, h} \mu(\theta_{-i}, h \mid I_i) \, u_i(a_i', \sigma_{-i}, \theta),

where \mu(\cdot \mid I_i) denotes player i's beliefs over the types of other players \theta_{-i} and histories h conditional on reaching I_i, u_i is player i's utility function, and the summation is over relevant types and histories consistent with I_i. This optimization ensures that no player has an incentive to deviate unilaterally at any information set, given the continuation strategies and beliefs.^[10]^[11] The primary implication of sequential rationality is the elimination of non-credible threats or deviations, as it requires strategies to be robust to local perturbations at every node, refining coarser solution concepts like Bayesian Nash equilibrium by preventing equilibria supported only by implausible off-path behavior. In games with incomplete information, this condition guarantees that players act as if solving a Bayesian game at each information set, promoting credibility in dynamic interactions.^[12] In complete information games, sequential rationality is analogous to subgame perfection, where strategies must be Nash equilibria in every subgame; the PBE extension adapts this idea to incomplete information settings by incorporating beliefs to handle non-singleton information sets, ensuring optimality in belief-contingent continuation games.^[1]

Consistency via Bayes' Rule

In a perfect Bayesian equilibrium, the consistency condition requires that players' beliefs about the state of the world or opponents' types at any information set be formed and updated using Bayesian inference based on the prior distribution and the equilibrium strategies, whenever the relevant history is reachable with positive probability. Specifically, for a history h with positive probability \Pr(h) > 0, the posterior belief \mu(\theta | h) over types \theta is given by Bayes' rule:

\mu(\theta | h) = \frac{\pi(\theta) \Pr(h | \theta)}{\Pr(h)},

where \pi(\theta) denotes the common prior over types, and \Pr(h | \theta) is the probability of observing history h conditional on type \theta under the equilibrium strategies.^[1] This ensures that beliefs along the equilibrium path (on-path beliefs) are fully derived from the strategies and priors, maintaining coherence with the observed actions.^[1] For off-path histories where \Pr(h) = 0, the denominator in Bayes' rule is zero, rendering direct application impossible; in such cases, beliefs are specified to support sequential rationality, meaning they cannot contradict the equilibrium strategies in a way that would incentivize deviations. Off-path beliefs must also satisfy the "no-signaling-what-you-don't-know" condition, ensuring that beliefs about a player's type, given another player's type and history, are independent of actions by players who do not possess information about that type.^[1] There are no further restrictions on these off-path beliefs beyond ensuring that the specified strategies remain optimal given them, allowing flexibility in belief assignment as long as it upholds the equilibrium.^[1] Overall, consistency in perfect Bayesian equilibrium demands that beliefs be derived from the prior and strategies wherever \Pr(h) > 0, providing a Bayesian foundation for decision-making at every stage.^[1] However, the freedom in specifying off-path beliefs can lead to equilibria sustained by implausible assessments, a limitation addressed in refinements like sequential equilibrium, which impose additional consistency requirements even off the path.^[1]

Refinement of Bayesian Nash Equilibrium

A Bayesian Nash equilibrium (BNE) is a strategy profile in which, for each player and each of their possible types, the strategy maximizes the player's expected utility given the strategies of the other players and the player's prior beliefs about the types of the others.^[13] This concept extends the standard Nash equilibrium to games of incomplete information by incorporating type-contingent strategies and Bayesian updating based on priors, but it does not account for the sequential nature of moves in dynamic settings.^[13] Perfect Bayesian equilibrium (PBE) refines BNE by imposing sequential rationality at every information set, ensuring that players' strategies are optimal given their beliefs about others' types not only on the equilibrium path but also off the equilibrium path.^[5] In addition to the best-response condition of BNE, PBE requires that beliefs be derived from Bayes' rule whenever possible on the equilibrium path and that strategies remain optimal given those beliefs everywhere in the game tree.^[5] This addresses a key shortcoming of BNE in sequential games, where equilibria may feature "empty threats" or non-credible actions off the path because BNE only verifies optimality at reached information sets, potentially allowing strategies that would not be sustained if deviations occurred.^[5] A divergence between BNE and PBE arises in simple entry deterrence games, such as the model analyzed by Milgrom and Roberts, where an incumbent firm of unknown strength (strong or weak type) faces a potential entrant.^[14] In a BNE, deterrence can be sustained by having the entrant stay out, the incumbent commit to fighting any entry regardless of type, and off-path beliefs after entry assigning probability 1 to the incumbent being strong, making entry unprofitable for the entrant.^[14] However, this fails as a PBE because the weak-type incumbent would not rationally fight at the post-entry information set—accommodation is optimal for the weak type given the entrant's action—violating sequential rationality unless beliefs are adjusted to make the fight credible, which they cannot be without contradicting consistency requirements.^[14] Thus, PBE eliminates such deterrence equilibria that rely on unreasonable off-path beliefs. In one-shot games of incomplete information, where there are no proper subgames or unreached information sets requiring off-path optimization, PBE coincides with BNE, as the sequential rationality condition imposes no additional restrictions beyond the standard best-response requirement.^[5]

Relation to Sequential Equilibrium

Perfect Bayesian equilibrium (PBE) and sequential equilibrium both refine the concept of Bayesian Nash equilibrium by incorporating sequential rationality in dynamic games with incomplete information, ensuring that strategies are optimal at every information set given players' beliefs.^[10] Sequential equilibrium, introduced by Kreps and Wilson in 1982, provides a framework for these refinements in games of incomplete information.^[6] Specifically, a sequential equilibrium consists of a strategy profile and a belief system that are the limits of sequences of PBEs, where each approximating PBE arises from a sequence of totally mixed strategies (with positive probabilities on all actions) that converge to the equilibrium strategies, and beliefs are updated via Bayes' rule from these fully mixed approximations.^[6] This trembling-hand approach ensures that beliefs at every information set, including those off the equilibrium path, are derived from plausible perturbations of the equilibrium, avoiding arbitrary specifications.^[1] A key difference lies in the treatment of off-equilibrium-path beliefs. In PBE, beliefs at information sets not reached under the equilibrium strategies can be specified freely, as long as they are consistent with Bayes' rule on the equilibrium path and satisfy sequential rationality; this flexibility often leads to multiple possible PBEs for the same game, as off-path beliefs can be chosen ad hoc to support particular deviations.^[15] Sequential equilibrium resolves this incompleteness by restricting off-path beliefs through the trembling-hand perfection implicit in the limiting process: even improbable "trembles" (mistakes) must be incorporated into belief formation, ensuring that only beliefs robust to small perturbations qualify as part of the equilibrium.^[6] As a result, sequential equilibrium selects a proper subset of all PBEs, eliminating those supported solely by unreasonable or non-robust off-path beliefs.^[15] Sequential equilibrium was developed by Kreps and Wilson in 1982 to extend subgame perfection to games of incomplete information. PBE, introduced later in 1991 by Fudenberg and Tirole, provides a foundational structure for analyzing signaling and dynamic games that is less restrictive than sequential equilibrium, while the latter's additional consistency criteria ensure greater robustness against coordinated deviations and belief manipulations.^[10]^[5]^[6]

Illustrative Examples

Signaling Games: Gift Giving

In the gift-giving signaling game, the sender possesses private information about her type, which is high (altruistic or high-value partner) with probability p or low with probability $1-p. The sender chooses the gift size, either small or large, incurring a cost that is lower for the high type due to greater altruism. The receiver observes the gift size, updates her beliefs about the sender's type using Bayes' rule where possible, and then decides whether to accept or reject, where acceptance may initiate a trade or relationship with payoffs reflecting the inferred type's value. Sender payoffs include the negative cost of the gift minus any rejection penalty, plus benefits from acceptance modulated by altruism; receiver payoffs are the value of the relationship if accepted (higher for high type) or zero if rejected or no gift sent.^[16]^[17] Perfect Bayesian equilibria in this game require strategies and beliefs satisfying sequential rationality at every information set and consistency of beliefs with Bayes' rule on the equilibrium path. A separating equilibrium arises when the high type sends a large gift and the low type sends a small gift. On the equilibrium path, the receiver's belief upon observing a large gift is that the sender is high type with probability 1, leading her to accept (as the expected value exceeds the rejection payoff of 0); upon observing a small gift, the belief is low type with probability 1, and she rejects if the low-type value is negative or sufficiently low. Off-equilibrium path, beliefs can be specified to deter deviations, such as assigning probability 0 to high type for unexpected gift sizes. This equilibrium is sequentially rational for the sender, as the high type's net benefit from acceptance outweighs the large gift cost, while the low type prefers the small gift (or no acceptance) over mimicking at higher cost without gain.^[17]^[18] Pooling equilibria occur when both types send the same gift size, leaving the receiver's on-path belief at the prior p. For instance, both types sending a small gift is a pooling PBE if the receiver accepts when p makes the expected value positive, with off-path beliefs for a large gift specifying low probability of high type to induce rejection and deter deviation. General belief consistency requires that off-path beliefs are arbitrary but must support sequential rationality.^[16] In Gift Game 1, the receiver's acceptance payoff is +1 for high type and -1 for low type (with rejection yielding 0), so acceptance is optimal only if the belief in high type exceeds 1/2. A low-type pooling equilibrium—both types send no (or small) gift, yielding payoffs (0,0)—is supported when p < 1/2, with pessimistic off-path beliefs that a large gift signals low type (belief ≤ 1/2), prompting rejection and preventing high-type deviation. Equilibrium payoffs are thus (0,0) for both players. Sequential rationality holds at the receiver's information sets: reject large gift under off-path beliefs, and the game ends without reaching acceptance sets for small gifts.^[18]^[16] In Gift Game 2, the receiver's acceptance payoff is positive regardless of type (e.g., +1 for both, rejection 0), so she always accepts any observed gift. A separating PBE exists where the high type sends a large gift (accepted, belief 1 for high) and the low type sends small (accepted, but net lower due to costs); intuitive off-path beliefs assign high type to large gifts and low to small. Multiple PBEs arise, including pooling on large gifts (both accepted, belief p) if costs allow, contrasting with Gift Game 1 by eliminating rejection as a disciplining device and relying on cost differences for separation. Verification confirms sequential rationality: the receiver accepts at all information sets reached after a gift, and sender strategies optimize given certain acceptance.^[18]

Dynamic Games: Jump Bidding

In dynamic games such as sequential auctions, jump bidding serves as a strategic tool under incomplete information, where bidders have private valuations for the object being auctioned. Consider a two-bidder English auction with affiliated values drawn from a common distribution, starting from an initial price p_0 and requiring bids to increase by at least a minimum increment m. Bidders alternate in submitting bids, and a jump bid occurs when a player bids substantially higher than the current price plus m, effectively skipping intermediate levels to signal strength. This setup allows the high-valuation bidder to intimidate the opponent, deterring further competition while maintaining the auction's sequential nature.^[19] In perfect Bayesian equilibrium (PBE), the high-valuation type employs a jump-bid strategy early in the auction to credibly signal its strength, prompting the opponent to update beliefs about the jumper's value distribution. The equilibrium strategy specifies that only sufficiently high types jump bid at certain price thresholds, while low types follow the minimum increment. Upon observing a jump, the non-jumping bidder infers that the jumper's value exceeds the jumped-to price with a probability updated via Bayes' rule from the prior distribution F(v) and the equilibrium bidding functions, concentrating beliefs on higher valuations. Off-equilibrium path, if a low type were to jump, beliefs are specified to attribute it to a high type, preventing profitable deviations like underbidding.^[19] Sequential rationality ensures that, at every information set, players' strategies are optimal given updated beliefs. In the second stage following a jump, the non-jumper rationally drops out if the expected value conditional on the updated beliefs falls below the current price, yielding no profitable deviation for either player. This equilibrium structure, solved via backward induction, sustains the jump bid as a best response, as the high type benefits from the opponent's withdrawal more than the cost of overbidding.^[19] The outcome of this PBE is that jump bidding acts as a credible signal of high valuation, leading to more efficient allocations by reducing the likelihood of the low-value bidder winning, compared to static auctions without such signaling opportunities. This refinement over straightforward bidding enhances the auction's performance under incomplete information, akin to signaling in simpler games but adapted to multi-move sequences.^[19]

Repeated Games: Public Goods Contribution

In repeated public goods games with incomplete information, players engage in an infinitely repeated stage game where each period involves a decision to contribute to a shared public good or defect by withholding contribution. Each player has a fixed private type—either a "normal" or cooperative type that values mutual contribution or a "greedy" or defector type that prefers unilateral defection—drawn independently at the outset with known prior probabilities. Contributions are publicly observed each period, enabling opponents to update their beliefs about a player's type based on the history of actions, while payoffs are discounted by a common factor \delta < 1. This setup introduces reputation dynamics, as observed behavior influences future interactions in a way that can sustain higher contributions than in the one-shot game.^[20] Perfect Bayesian equilibria (PBEs) in this framework leverage reputation effects to encourage contributions from rational players who might otherwise defect. In equilibrium strategies, players with uncertain types (from opponents' perspectives) contribute to mimic the behavior of a cooperative type, thereby building or preserving beliefs that they are likely cooperators and avoiding punishment. These strategies often take the form of trigger mechanisms conditioned on the full history of contributions, such as cooperating as long as no defections are observed and shifting to punishment (non-contribution) following deviations. For instance, in war-of-attrition-style PBEs, players delay or alternate contributions to reveal types gradually, with rational defectors occasionally conceding to maintain reputation. Such constructions ensure that short-term defection costs future cooperation, making sustained contribution incentive-compatible.^[20] Beliefs in these PBEs evolve through Bayesian updating on the sequence of observed contribution histories, refining the posterior probability that a player is a cooperator versus defector. On the equilibrium path, consistent high contributions reinforce cooperative beliefs, supporting ongoing mutual benefit. Off the equilibrium path, a low or zero contribution prompts a downward revision in beliefs toward the defector type, often leading to pessimistic punishment beliefs where opponents anticipate future non-cooperation and respond by defecting themselves. This updating process is consistent with Bayes' rule wherever possible, given the public observability of actions, and handles incomplete revelation by maintaining uncertainty over infinite horizons.^[20] Full cooperation—where all players contribute every period—can be sustained as a PBE when the discount factor \delta is sufficiently high, as the long-run value of preserved reputation outweighs the immediate gain from defection. In such equilibria, the stage-game payoffs approach the cooperative outcome, provided parameters like type probabilities and payoff costs satisfy incentive constraints. However, multiple PBEs exist, varying in the reputation-building costs; for example, some equilibria impose stricter triggers with harsher punishments, while others allow more leniency, leading to different payoff sets that converge as \delta \to 1. Notably, for certain parameter values (e.g., high defection gains), undiscounted versions (\delta = 1) may lack any Nash equilibrium, but limit PBEs from discounted games fill this gap with cooperative payoffs.^[20] The rationality of these PBEs is verified through sequential rationality at every information set and history, ensuring that each player's action maximizes their expected discounted payoff given the current beliefs about opponents' types and strategies. Deviation incentives are checked period-by-period, confirming that contributing (or punishing) is optimal under the evolving beliefs, while belief consistency prevents arbitrary off-path specifications. This refinement distinguishes PBEs from weaker Bayesian Nash equilibria by eliminating non-credible threats in the dynamic setting.^[20]

Applications and Extensions

In Economic Models

Perfect Bayesian equilibrium (PBE) plays a central role in analyzing adverse selection in insurance markets, where insurers cannot observe the risk types of potential customers. In the seminal Rothschild-Stiglitz model, high-risk individuals seek more coverage than low-risk ones, leading to separating equilibria where insurers offer contracts that screen types through self-selection, with off-equilibrium beliefs updated via Bayes' rule to deter deviations.^[21] These equilibria are supported as PBEs, ensuring sequential rationality and consistent beliefs even when pooling contracts are considered but rejected based on the threat of cream-skimming by rivals.^[22] In principal-agent problems involving dynamic contracts, PBE addresses moral hazard when the agent's effort is unobservable, allowing the principal to update beliefs about the agent's type based on observed outputs over time. For instance, in repeated interactions, the principal designs incentive-compatible contracts that reward high outputs to induce effort, with PBE requiring that beliefs off the equilibrium path—such as after unexpectedly low outputs—remain consistent with Bayes' rule, preventing unraveling of the contract.^[23] This approach yields second-best allocations that balance rent extraction with incentive provision in incomplete information settings.^[23] PBE also explains market entry deterrence through predatory pricing as a signaling mechanism, where an incumbent with private information about low costs sets temporarily low prices to convince potential entrants of its ability to sustain competition. In Milgrom and Roberts' signaling model, the incumbent's aggressive pricing signals low marginal costs, updating entrants' beliefs to expect lower post-entry profits due to tougher competition, thus deterring inefficient entry in equilibrium. This PBE outcome highlights how predation can be rational under asymmetric information, though it may reduce social welfare by blocking viable entrants.^[24] In policy design for incomplete information environments, PBE informs regulatory applications in auction theory, particularly during the 1990s spectrum auctions where mechanisms were crafted to elicit truthful bidding under Bayesian updating of valuations. Milgrom and Wilson's analysis of common-value auctions demonstrates how PBE refines bidding strategies, guiding designs like the simultaneous multiple round auction to mitigate the winner's curse and ensure efficient allocation.^[25] These insights influenced U.S. FCC policies, promoting revenue maximization while accommodating bidders' private information.^[26] Empirically, lab experiments testing PBE in signaling games, such as those by Brandts and Holt, show that subjects often converge to intuitive equilibria under equilibrium dominance, where implausible off-path beliefs are eliminated, supporting the predictive power of PBE in adverse selection scenarios.^[27]

Computational and Empirical Analysis

Computing Perfect Bayesian Equilibria (PBEs) in finite-horizon games typically relies on backward induction, starting from terminal nodes and working backwards to derive optimal strategies and consistent beliefs at each information set, ensuring sequential rationality throughout the game tree.^[28] For infinite-horizon games, methods such as value iteration or forward recursive algorithms approximate PBEs by solving fixed-point equations that incorporate discounting and belief updates over time, often decomposing the problem into structured belief-state spaces.^[29] Software tools like Gambit facilitate these computations by modeling extensive-form games with incomplete information and enumerating equilibria, including PBEs, through recursive solving of reduced normal forms or polynomial systems. Recent algorithms compute approximate perfect Bayesian equilibria in sequential auctions with continuous action and value spaces, improving tractability for realistic models.^[30] A key challenge in computing PBEs arises from the multiplicity of equilibria, particularly due to the flexibility in specifying off-path beliefs at unreached information sets, which can support a continuum of consistent belief profiles without violating Bayes' rule on equilibrium paths.^[28] To address this, selection criteria such as trembling-hand perfection are applied, where equilibria are limits of sequences of perturbed strategies that approximate small implementation errors, thereby restricting off-path beliefs to those robust to minor deviations.^[10] Empirical validation of PBEs often employs structural estimation techniques in auction settings, where researchers invert observed bidding data to recover private value distributions under the assumption of Bayesian Nash equilibrium strategies, as in first-price auctions analyzed by Bajari and Hortaçsu in the early 2000s.^[31] Laboratory experiments reveal deviations from PBE predictions, particularly in belief updating, with post-2010 studies showing non-Bayesian overweighting of confirmatory evidence in social learning tasks, leading to behavioral critiques of the model's assumption of rational Bayesian inference.^[32] Recent advances integrate artificial intelligence, using reinforcement learning algorithms like actor-critic methods to approximate PBEs in complex imperfect-information games such as multi-player poker, where deep neural networks learn near-equilibrium policies by iteratively updating value functions and beliefs in no-limit Hold'em variants during the 2020s.^[33] Despite these developments, PBEs remain sensitive to arbitrary off-path beliefs, which can drastically alter equilibrium outcomes, prompting refinements through universal type spaces that expand the type set to encompass all possible higher-order beliefs for robustness.^[10]

References

[1]
[PDF] Perfect Bayesian and sequential equilibria : a clarifying note
PERFECT BAYESIAN AND SEQUENTIAL EQUILIBRIA: A CLARIFYING NOTE. By. Drew Fudenberg and Jean Tirole. No. 496. May 1988 massachusetts institute of technology. 50 ...
[2]
[PDF] MS&E 246: Lecture 15 Perfect Bayesian equilibrium
In this lecture, we begin a study of dynamic games of incomplete information. We will develop an analog of Bayesian equilibrium for this setting, called perfect ...
[3]
[PDF] A General, Practicable Definition of Perfect Bayesian Equilibrium
This paper develops a general definition of perfect Bayesian equilibrium (PBE) for extensive-form games. It is based on a new consistency condition for the ...
[4]
games with incomplete information played by "bayesian" players - jstor
Part II of the paper will now show that any Nash equilibrium point of this Bayes- ian game yields a "Bayesian equilibrium point" for the original game and ...
[5]
Sequential Equilibria - jstor
Sequential equilibria require players' strategies to be sequentially rational, where every decision is part of an optimal strategy for the remainder of the ...
[6]
Signaling Games and Stable Equilibria - jstor
An important recent example of this is the Kohlberg-Mertens [1986] theory of stability and stable equilibrium outcomes. Because the Kohlberg-Mertens development ...
[7]
Perfect Bayesian equilibrium and sequential equilibrium
We introduce a formal definition of perfect Bayesian equilibrium (PBE) for multi-period games with observed actions.<|control11|><|separator|>
[8]
[PDF] Behavioral Game Theory Experiments and Modeling
Apr 23, 2012 · Many behavioral game theory models also circumvent two long-standing problems in equilibrium game theory: Refinement and selection. These models ...
[9]
Computing Perfect Bayesian Equilibria in Sequential Auctions with ...
Apr 11, 2025 · We present an algorithm for computing pure-strategy epsilon-perfect Bayesian equilibria in sequential auctions with continuous action and value spaces.
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[PDF] Perfect Bayesian Equilibrium: Consistency Conditions for Practical ...
Fudenberg and Tirole (1991) provides the focal PBE definition for a class of games known as finite multi-stage games with observed actions and independent types ...
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[PDF] Perfect Bayesian Equilibrium When players move sequentially and ...
In a separating equilibrium, player 1's types choose dif- ferent actions, so player 2 will be able to infer player 1's type by observing his action.
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[PDF] 14.12 Game Theory Lecture Notes Lectures 15-18 - MIT
Definition 5 A Perfect Bayesian Nash Equilibrium is a pair (s,b) of strategy profile. and a set of beliefs such that. 1.
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Games with Incomplete Information Played by “Bayesian” Players, I ...
Games with Incomplete Information Played by “Bayesian” Players, I–III Part I. The Basic Model. John C. Harsanyi.
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Predation, reputation, and entry deterrence - ScienceDirect.com
Our gametheoretic, equilibrium analysis suggests that if a firm is threatened by several potential entrants, then predation may be rational against early ...
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Exploring the Gap between Perfect Bayesian Equilibrium and ...
In this paper we explore the gap between PBE and sequential equilibrium, by identifying solution concepts that lie strictly between PBE and sequential ...Missing: via seminal
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### Summary of the Gift Game from http://spia.uga.edu/faculty_pages/dougherk/svt_10_bayesian_subgame_perfect.pdf
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[PDF] Game Theory: Perfect Equilibria in Extensive Form Games
May 15, 2008 · For example, in the gift-giving game, if player 2 thinks player 1 is playing the strategy NG, then if player 2 ever observes a gift offer, she ...
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[PDF] Perfect Bayesian Equilibrium When players move sequentially and ...
When players move sequentially and have private infor- mation, some of the Bayesian Nash equilibria may involve strategies that are not sequentially rational.Missing: seminal paper
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Strategic Jump Bidding in English Auctions - Oxford Academic
This paper solves for equilibria of sequential bid (or English) auctions with affiliated values when jump bidding strategies may be employed to intimidate one's ...
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https://doi.org/10.1016/j.jet.2015.05.014
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[PDF] Equilibrium in Competitive Insurance Markets
Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect. Information. Author(s): Michael Rothschild and Joseph Stiglitz. Source ...Missing: Bayesian | Show results with:Bayesian
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[PDF] Characterization and Uniqueness of Equilibrium in Competitive ...
Nov 5, 2014 · This paper studies subgame perfect equilibrium in the competitive insurance model of Rothschild and Stiglitz (1976).
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[PDF] The Informed Principal with Agent Moral Hazard
Sep 26, 2022 · The key outcomes are the “Rothschild-Stiglitz-Wilson (RSW) allocations.” An allocation is a perfect Bayesian equilibrium outcome if and only if ...
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A Signaling Model of Predatory Pricing - jstor
predatory behavior is not benign, because the rational anticipation of low prices may deter entry which would be socially desirable and would occur if predation ...
[25]
[PDF] Improvements to auction theory and inventions of new auction formats
Oct 12, 2020 · Wilson (1969) provided the first equilibrium analysis of optimal bidding under common values when he derived Bayesian Nash equilibria in his ...
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[PDF] The Unity of Auction Theory: Milgrom's Masterclass
Dec 4, 2024 · In chapter one, he shows that the 1990 New Zealand spectrum auction's failure to raise the revenue antici- pated can be traced to its seriously ...
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An Experimental Test of Equilibrium Dominance in Signaling Games
This paper surveys experimental studies of sender-receiver games. In these games a sender, with some private information, selects a signal that is observed ...
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[PDF] Game Theory, Lecture 2: Equilibrium Refinements
Backward induction theorem: every finite extensive-form game of perfect information (i.e., all singleton info sets) has a pure-strategy SPE, which can be found ...
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Structured Perfect Bayesian Equilibrium in Infinite Horizon Dynamic ...
Sep 14, 2016 · In this paper we find a subset of PBE for an infinite horizon discounted reward asymmetric information dynamic game.Missing: iteration | Show results with:iteration
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[PDF] NBER WORKING PAPER SERIES ARE STRUCTURAL ESTIMATES ...
These papers all assume that bidders are playing a Bayes-Nash equilibrium to the auction game, and es- timate the distribution of bidders' private information ...
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[PDF] Non-Bayesian updating in a social learning experiment
We found an important deviation from Bayesian updating: news contradicting the previous belief are overweighted compared to news that confirm it. This ...<|separator|>
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Optimal Policy of Multiplayer Poker via Actor-Critic Reinforcement ...
This paper proposes an optimal policy learning method for multi-player poker games based on Actor-Critic reinforcement learning.