Game Theory 11 – chap21. Forward induction and Cho-Kreps’ intuitive criterion

5. Dynamic Bayesian Games
5.3 Dynamic Bayesian Games: Signaling
– Forward Induction and Cho-Kreps’ Intuitive Criterion*

Forward Induction: Motivation and Ideas

As in the Beer-Quiche game, most signaling games posses multiple PBE

EXAMPLE 5.2 (BATTLE OF THE SEXES WITH AN OUTSIDE OPTION).

There exist two pure-strategy PBE in this game:

• ⟨(CO, o), \(\mu_2(x)\) = 1⟩
• ⟨(NM, m), \(\mu_2(x)\) ≤ 3/4⟩

(∵ p ≤ 3(1-p)) ⇔ p ≤ 3/4)

Also two pure-strategy SPE:
(CO, o) and (NM, m)

Are they all “reasonable” predictions?

KOHLBERG AND MERTENS (1986) use the previous example to introduce the idea of forward induction:

“A subgame should not be treated as a separate game, because it was preceded by a very specific form of preplay communication. In the previous example, it is common knowledge that, when player 2 has to play in the subgame, preplay communication has effectively ended with the following message from player 1 to player 2:

‘Look, I had an opportunity to get 2 for sure,
and nevertheless I decided to play in this subgame.’“

Backward induction:
a predecessor reasons about what will rationally happen at future points of the game

Forward induction:
a successor reasons about what could’ve rationally happened at previous points of the game

Refinements of PBE with Belief Restrictions

We now discuss how to sort out PBE with beliefs that are inconsistent with forward induction

A weak form of forward induction rules out equilibria that vanish after any dominated strategies are removed

EXAMPLE 5.3.

Suppose message I was sent

Message I is strictly dominated for H types

So player 2 ought to believe he is facing L:

\(\mu_2^I(H) = 0, \mu_2^I(L) = 1\)

Under this belief player 2 would play D

Anticipating player 2’s behavior, the type-H would send message O and type-L would send I, leading us to a PBE based on reasonable beliefs:

⟨(OI, D), \(\mu_2^I\)(L) = 1⟩.

There exists another PBE where player 2 responds by U under the Bayesian (but not reasonable) beliefs \(\mu_2^I\)(L) ≤ 0.5 and both types of player 1 send message O.

⟨(OO, U), \(\mu_2^I\)(L) ≤ 0.5⟩

Strict dominance (a weak form of forward induction) eliminates this PBE

Another (strong) form of forward induction is equilibrium dominance:
It rules out eqba that vanish after strategies that are not best response at the given eqbm are removed.

Recall that the Beer-Quiche game has two pooling equilibria:

⟨(BB, WF), p = 0.1, q ≥ 0.5⟩ ⇒ ⟨(BB, WF), p = 0.1, q = 1⟩

The other pooling equilibrium was the next where message “B” is unused.

⟨(QQ, FW), p ≥ 0.5, q = 0.1⟩

In the weak type, player 2 does not order Beer because he gets a payoff of 3 from Q
(in the weak type, Beer is never the best response)

We can determine that player 1 has a strong type when he orders a Beer, because a strong type allows him to get a higher payoff of 3.
Hence 1-p = 1 ⇔ p = 0

By the following formula, both player 1 and 2 debate.
So this equilibrium can be erased by the intuitive criterion of Cho-Kreps, so it does not hold.

⟨(QB, WW), p = 0, q = 0.1⟩

We say that this PBE fails the Cho-Kreps criterion (or the intuitive criterion)

Strict vs Eqbm Dominance

In Example 5.3, certain beliefs are deemed unreasonable because they are based on expecting a particular sender type to play a dominated strategy

This is not the case in the Beer-Quiche game: neither B nor Q is strictly dominated

Instead, we fix the component of equilibria and examine if certain beliefs are unreasonable given the anticipation of equilibrium payoffs

The possible payoffs from B to the weak type are always smaller than the equilibrium payoff from Q, so if player 2 receives message B, player 2 should not think he is facing the weak type

• Reference: Chang-Koo Chi, (39/50) Game Theory and Applications 11 – Forward induction and Cho-Kreps’ intuitive criterion, Jul 15, 2020, https://youtu.be/chxRL9GlHxg