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A company offers you a free trial. It might be a good product hoping you will love it, or a lock-in hoping you will get stuck. You cannot tell which from the offer alone. Whether you should accept depends entirely on what a free trial makes you believe.

Perfect Bayesian equilibrium

When players move in turn and some of them hide private information, equilibrium needs two things at once: a strategy for everyone, and a belief at every point about what has happened so far. Perfect Bayesian equilibrium ties them together, so each move is optimal given beliefs, and beliefs are updated from the moves.

From earlier lessons

Two quick ones before we start.

In a Bayesian game you best-respond to...

Subgame perfection requires optimal play...

Predict, then play

The Gift game. A sender is a Friend with probability $p$ , or an Enemy otherwise, and chooses whether to give you a gift. If you accept, you receive $+1$ from a Friend’s gift and $-1$ from an Enemy’s; rejecting gives you $0$ . A sender whose gift is accepted receives $+1$ , a rejected sender receives $-1$ , and not giving yields $0$ . Set the prior and check whether “both types give” can hold up.

Why did that happen?

Take a guess, then reveal the reason

Your decision hinges on a belief: how likely is this gift to come from a Friend? If both types give, the gift is uninformative, so your belief stays at the prior. When the prior is high enough you accept, and giving pays off for both types, so the arrangement holds. When the prior is too low you reject, and a gift becomes a trap you avoid, so no one bothers to give. The equilibrium is a strategy and a belief that agree with each other.

The math

Definition, perfect Bayesian equilibrium

A perfect Bayesian equilibrium is a strategy profile together with a belief at every information set such that: (1) beliefs on the equilibrium path are derived from the strategies by Bayes’ rule; (2) every player acts sequentially rationally, choosing the best action given their belief at each information set, on and off the path; and (3) off-path beliefs are unrestricted by Bayes but the actions they induce must still be optimal. It extends subgame perfection to games where private information means proper subgames may not exist.

In the Gift game the receiver’s belief after a gift is $b = \Pr(\text{Friend} \mid \text{gift})$ . Accepting is worth $b\cdot 1 + (1-b)\cdot(-1) = 2b - 1$ , and rejecting is worth $0$ , so the receiver accepts iff $b \ge \tfrac12$ .

Consider both types giving. On the path Bayes sets $b = p$ , the prior. If $p \ge \tfrac12$ the receiver accepts, and since both senders prefer a gift accepted (payoff 1) to not giving (0), pooling on Give is a PBE. If $p < \tfrac12$ the receiver rejects, both senders then earn $-1 < 0$ and deviate to Not, so pooling on Give fails. Pooling on Not can be a PBE when an off-path gift induces a sufficiently pessimistic belief. This is one of several equilibria the full signaling game may admit under different strategies and off-path beliefs.

Work it out

Two priors, two equilibria

p = 0.6. Suppose both types give. On the path a gift implies belief $0.6$ , and accepting is worth $2(0.6) - 1 = 0.2 > 0$ , so the receiver accepts. Both senders then get $1$ from giving versus $0$ from not, so neither deviates. Pooling on Give is a PBE.

p = 0.4. Suppose both types give. Now a gift implies belief $0.4$ , accepting is worth $-0.2 < 0$ , so the receiver rejects, and both senders get $-1 < 0$ : they deviate to Not. The surviving PBE is pooling on Not. To support it, assign the off-path belief after a (never-sent) gift to be Enemy, so a deviating giver would be rejected and earn $-1$ , confirming that not giving is optimal.

Key idea

A perfect Bayesian equilibrium pairs strategies with beliefs: act optimally given your belief at every point, and update beliefs by Bayes wherever you can. A signal is trusted only when the belief it induces clears the bar for acting on it.

Check yourself

Check yourselfapply

Both types give a gift. What belief does the receiver hold after seeing one, and why?

Check yourselfanalyse

At p = 0.4, pooling on Not is the equilibrium. What role do off-path beliefs play?

Check yourselfbrings back: subgame perfection

Why do we need PBE rather than just subgame perfection for the Gift game?

Sources

Fudenberg, D., and Tirole, J. (1991). Perfect Bayesian equilibrium and sequential equilibrium. Journal of Economic Theory, 53(2), 236-260.