![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/\uc2a4\ud06c\ub9b0\uc0f7-2023-07-06-\uc624\ud6c4-8.52.53.png\")
<\/figure><\/div>\n\n\nThere exist two pure-strategy PBE in this game:<\/p>\n\n\n\n
- \n
- \u27e8(CO, o), \\(\\mu_2(x)\\) = 1\u27e9<\/li>\n\n\n\n
- \u27e8(NM, m), \\(\\mu_2(x)\\) \u2264 3\/4\u27e9<\/li>\n<\/ul>\n\n\n\n
(\u2235 p \u2264 3(1-p)) \u21d4 p \u2264 3\/4)<\/mark><\/p>\n\n\n\n
Also two pure-strategy SPE:
(CO, o) and (NM, m)<\/p>\n\n\n\nAre they all “reasonable” predictions?<\/p>\n\n\n\n
<\/p>\n\n\n\n
KOHLBERG AND MERTENS (1986) use the previous example to introduce the idea of forward induction<\/mark><\/strong>:<\/p>\n\n\n\n
“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:<\/p>\n\n\n\n
‘Look, I had an opportunity to get 2 for sure,
and nevertheless I decided to play in this subgame.’<\/em>“<\/p>\n\n\n\nBackward<\/mark><\/strong> induction:
a predecessor reasons about what will rationally happen at future<\/mark><\/strong> points of the game<\/p>\n\n\n\nForward<\/mark><\/strong> induction:
a successor reasons about what could’ve rationally happened at previous<\/mark><\/strong> points of the game<\/p>\n\n\n\nRefinements of PBE with Belief Restrictions<\/h3>\n\n\n\n
We now discuss how to sort out PBE with beliefs that are inconsistent<\/em> with forward induction<\/p>\n\n\n\n
A weak form of forward induction rules out equilibria that vanish after any dominated<\/em> strategies are removed<\/p>\n\n\n\n
EXAMPLE 5.3<\/mark>.<\/p>\n\n\n
\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/image-46.png\")
<\/figure><\/div>\n\n\nSuppose message I was sent<\/p>\n\n\n\n
Message I is strictly dominated for H types<\/p>\n\n\n\n
So player 2 ought to believe he is facing L:<\/p>\n\n\n\n
\\(\\mu_2^I(H) = 0, \\mu_2^I(L) = 1\\)<\/mark><\/p>\n\n\n\n
Under this belief player 2 would play D<\/p>\n\n\n\n
<\/p>\n\n\n\n
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<\/em> beliefs:<\/p>\n\n\n\n
\u27e8(OI, D), \\(\\mu_2^I\\)(L) = 1\u27e9.<\/p>\n\n\n\n
There exists another PBE where player 2 responds by U under the Bayesian (but not reasonable<\/mark><\/strong>) beliefs \\(\\mu_2^I\\)(L) \u2264 0.5 and both types of player 1 send message O.<\/p>\n\n\n\n
\u27e8(OO, U), \\(\\mu_2^I\\)(L) \u2264 0.5\u27e9<\/p>\n\n\n\n
Strict dominance<\/em> (a weak form of forward induction) eliminates this PBE<\/p>\n\n\n\n
<\/p>\n\n\n\n
Another (strong) form of forward induction is equilibrium dominance<\/em>:
It rules out eqba that vanish after strategies that are not best response at the given eqbm are removed.<\/p>\n\n\n\nRecall that the Beer-Quiche game has two pooling equilibria:<\/p>\n\n\n
\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/07\/image-14-1024x833.png\")
<\/figure><\/div>\n\n\n\u27e8(BB, WF), p = 0.1, q \u2265 0.5\u27e9 \u21d2 \u27e8(BB, WF), p = 0.1, q = 1<\/strong>\u27e9<\/p>\n\n\n\n
<\/p>\n\n\n\n
The other pooling equilibrium was the next where message “B” is unused.<\/p>\n\n\n\n
\u27e8(QQ, FW), p \u2265 0.5, q = 0.1\u27e9<\/p>\n\n\n\n
In the weak type, player 2 does not order Beer because he gets a payoff of 3 from Q<\/mark>
(in the weak type, Beer is never the best response)<\/mark><\/p>\n\n\n\nWe 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 \u21d4 p = 0<\/mark><\/p>\n\n\n\nBy 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.<\/mark><\/p>\n\n\n\n\u27e8(QB, WW), p = 0, q = 0.1\u27e9<\/mark><\/p>\n\n\n\n
We say that this PBE fails the Cho-Kreps criterion (or the intuitive criterion<\/em>)<\/p>\n\n\n\n
Strict vs Eqbm Dominance<\/h3>\n\n\n\n
In Example 5.3, certain beliefs are deemed unreasonable because they are based on expecting a particular sender type to play a dominated strategy<\/p>\n\n\n\n
This is not the case in the Beer-Quiche game: neither B nor Q is strictly dominated<\/p>\n\n\n\n
Instead, we fix the component of equilibria<\/mark><\/strong> and examine if certain beliefs are unreasonable given the anticipation of equilibrium payoffs<\/p>\n\n\n\n
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<\/p>\n\n\n\n
<\/p>\n\n\n\n