When players compete strategically, they must anticipate not only what their opponents will do on the path they expect the game to follow, but also what might happen if things go off track. In games with incomplete information, where players don’t know everything about their opponents, this becomes especially tricky. How should we form beliefs about situations that never actually occur in equilibrium? This question sits at the heart of understanding consistent assessments and off-the-path equilibrium in dynamic games.
Table of Contents
- The challenge of off-equilibrium beliefs
- The Kreps and Wilson consistency criterion
- Formalizing the consistency requirement
- Trembling hand perfect equilibrium
- Why trembles matter for strategic thinking
- Formalizing consistency with trembles
- The relationship between consistency and sequential rationality
- Practical implications for game analysis
The challenge of off-equilibrium beliefs
Perfect Bayesian equilibrium provides a framework for analyzing dynamic games with incomplete information. It requires that players act optimally given their beliefs, and that these beliefs are updated using Bayes’ rule wherever possible. But here’s the catch: what happens at information sets that are never reached along the equilibrium path?
Imagine a job market where employers try to infer a worker’s ability from their education choice. In equilibrium, high-ability workers might choose advanced degrees while low-ability workers don’t. But what should an employer believe if they encounter someone with a level of education that nobody chooses in equilibrium? The basic definition of perfect Bayesian equilibrium leaves these off-the-path beliefs unconstrained, allowing any beliefs whatsoever. This freedom can lead to implausible equilibria that rely on unreasonable threats or expectations.
The Kreps and Wilson consistency criterion
To address these problems, economists David Kreps and Robert Wilson proposed a consistency criterion that tightens the requirements for equilibrium beliefs. Their insight was elegant: even off-the-path beliefs should be justifiable as arising from some plausible process.
An assessment combining strategies and beliefs is consistent if it represents the limit of a sequence of assessments where strategies are completely mixed and beliefs are derived via Bayes’ rule. A completely mixed strategy means every action is played with some positive probability, even if that probability is very small. When all strategies are completely mixed, every information set is reached with positive probability, making Bayes’ rule applicable everywhere.
Think of it this way: consistency requires that your beliefs at unusual information sets should be close to what you’d calculate if you thought there was some tiny chance that players might choose any action. As these probabilities of unusual actions shrink toward zero, the beliefs should converge to something sensible. This prevents you from supporting an equilibrium with arbitrary, unreasonable beliefs just because certain situations never arise.
Formalizing the consistency requirement
More formally, consistency requires that players’ beliefs are close to those derived using Bayes’ rule for some sequence of small trembles in strategies, even for information sets off the equilibrium path. If we consider a sequence of perturbed games where each player assigns a small probability to every possible action, and if beliefs in these perturbed games are computed using Bayes’ rule, then the limiting beliefs as the perturbations vanish must equal the beliefs in the proposed equilibrium.
This applies to both on-path and off-path beliefs. On the equilibrium path, where information sets are reached with positive probability, consistency simply requires standard Bayesian updating. Off the path, however, consistency becomes a powerful restriction, ruling out beliefs that couldn’t arise as limits of sensible updating processes.
Trembling hand perfect equilibrium
The idea behind the Kreps-Wilson consistency criterion connects deeply to an earlier concept introduced by Reinhard Selten called trembling hand perfection. Selten’s insight was that players should allow for the possibility that opponents might make small mistakes or “trembles” in their play.
Imagine a chess player whose hand might slip and move a piece to an unintended square, or a negotiator who might accidentally reveal information they meant to conceal. These aren’t strategic choices but genuine errors that occur with negligible probability. The trembling hand concept ensures that every information set, even those far off the expected path of play, is reached with positive probability, allowing Bayes’ rule to be applied everywhere.
A strategy profile is trembling hand perfect if it’s the limit of equilibria in games where every action must be played with at least some small probability. This means each player’s strategy must be optimal not just against the exact equilibrium strategies of opponents, but also against slight perturbations of those strategies. If your strategy depends crucially on your opponent doing exactly one thing with certainty, it won’t be trembling hand perfect.
Why trembles matter for strategic thinking
The trembling hand idea does more than just handle off-path beliefs. It also eliminates weakly dominated strategies from equilibrium. A weakly dominated strategy is one that never does better, and sometimes does worse, than another available strategy. While Nash equilibrium might permit players to use weakly dominated strategies, trembling hand perfection rules them out.
Consider a simple example: you can choose between two actions, where action A gives you a payoff of 10 regardless of what your opponent does, while action B gives you 10 if your opponent does one thing and 5 if they do another. Action B is weakly dominated by action A. In a Nash equilibrium, you might play B if you’re certain your opponent will choose the action that makes B give you 10. But if there’s even the tiniest chance of a mistake, action A is strictly better. Trembling hand perfection enforces this reasoning.
Formalizing consistency with trembles
Bringing these ideas together, we can formalize consistency in terms of trembles. Consistency requires that for every information set in the game, including those off the equilibrium path, there must exist a sequence of completely mixed strategy profiles converging to the equilibrium strategies such that the beliefs at that information set converge to the equilibrium beliefs when computed via Bayes’ rule.
This requirement has bite in several ways. First, it ensures that beliefs respect the structure of the game tree and the relative likelihoods of different histories leading to an information set. Second, it prevents players from sustaining equilibria through incredible off-path beliefs. Third, it guarantees that the equilibrium concept refines both Nash equilibrium and subgame perfection.
The relationship between consistency and sequential rationality
An assessment that satisfies both consistency and sequential rationality is called a sequential equilibrium. Sequential rationality means that at every information set, a player’s strategy must maximize expected payoff given their beliefs. When combined with consistency, this creates a powerful equilibrium concept that eliminates many implausible equilibria while retaining existence properties.
The beauty of this approach is that it captures the intuition that strategic reasoning should be robust to small perturbations. If an equilibrium can only be sustained through beliefs that would immediately fall apart with the slightest uncertainty about behavior, it’s probably not a good prediction of what will happen. Consistency ensures that equilibrium beliefs could arise from a sensible learning or reasoning process, even in situations that don’t occur along the equilibrium path.
Practical implications for game analysis
These refinements matter greatly for applied game theory. In signaling games, where one player takes actions to reveal private information to another, weak perfect Bayesian equilibrium often admits too many equilibria. Some of these equilibria rest on off-path beliefs that seem arbitrary or implausible. By imposing consistency, analysts can narrow down predictions to more reasonable outcomes.
For instance, in the education signaling model mentioned earlier, consistency restrictions help determine what employers should believe when they observe unexpected education levels. Rather than allowing any beliefs whatsoever, consistency requires that these beliefs be close to what would emerge if there were small probabilities that workers of different abilities might choose unusual education levels. This often rules out pooling equilibria where all worker types choose the same action, supported only by pessimistic off-path beliefs.
Similarly, in bargaining or negotiation settings, consistency prevents parties from making threats that they wouldn’t actually want to carry out, even accounting for small chances of mistakes. This brings theoretical predictions closer to what we observe in real strategic interactions, where players do seem to account for the possibility that others might deviate from expected behavior.
What do you think? When you face strategic situations in business or everyday life, do you account for the possibility that others might make mistakes or act unpredictably? How might thinking about trembles and consistency change your approach to anticipating what others will do?
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