Game Theory 10 – chap17. Sequential rationality in games of imperfect information

5. Dynamic Bayesian Games
5.1 Games of Imperfect Information and Sequential Rationality*

The next example demonstrates that subgame perfection is dysfunctional in dynamic games of imperfect into (equivalently, incomplete info)

EXAMPLE 5.1. Player 1 first moves by choosing one of the three actions: T, M, and B. If player 1 chooses either T or M, then player 2 learns only that player 1 did not choose B, choosing between L and R.

1 \ 2	L	R
T	2, 1	0, 0
M	0, 2	0, 1
B	1, 3	1, 3

In (B, R) player 2’s behavior is not sequentially rational, because R is strictly dominated at his information set

Note that SPE = NE as there is no proper subgame

(T, L), (B, R) is NE

The next example demonstrates why we need another ingredient: player’s beliefs

EXAMPLE 5.2.

1 \ 2	L	R
T	4, 1	0, 0
M	3, 0	0, 1
B	2, 2	2, 2

In contrast with the previous example, neither L nor R is strictly dominated

Hence both strategies can be a best response, depending on player 2’s beliefs

Player 2’s BR at {x,y} = \(\begin{cases}L \quad p = Pr(\text{player 2 at x}) > \frac{1}{2} \\ \{L,R\} \quad p=\frac{1}{2} \\ R \quad p < \frac{1}{2}\end{cases}\)

Beliefs

Player i’s beliefs are a probability distribution over his decision nodes satisfying

\(\sum_{x \in I}\mu_i(x)\) = 1 for each information set \(I \subset D_i\).

e.g., in {a, b, c, d}, II = {{p, 1-p}, {q, 1-q}} = {{a, c}, {b, d}}
The sum of the decision nodes of a and c is 1, b and d are also 1, and beliefs can be defined by their respective probability distributions

EXAMPLE 5.3.

• No need to specify player 2’s beliefs
• Player 3 has two info sets,

\(\mathbb{I}_3\) = {{a, b}, {c}}.

• Player 3’s possible beliefs are

\(\mu_3 = (\mu_3(a), \mu_3(b))\)
\(\mu_3(a) + \mu_3(b) = 1\)

Bayesian Beliefs

Consider a decision node x in information set I

Given a strategy profile \(\sigma = (\sigma_1, \sigma_2)\), we define

\(P_\sigma(x)\) = the probability of node x being reached under \(\sigma\)
\(P_\sigma(I)\) = the probability of info set I being reached under \(\sigma\)

Player i‘s beliefs \(\mu\) are said to be Bayesian given \(\sigma\) if for every x ∈ I.

\(\mu_i(x) = \frac{P_\sigma(x)}{P_\sigma(I)}\) whenever \(P_\sigma(I)\) > 0

EXAMPLE 5.4. Suppose that the 3 players play according to \(\sigma\) in the following game:

\(P_sigma(x)\) = (0.5T + 0.5B, (0.2L + 0.8R, 0.8L + 0.2R), W)

\(P_\sigma(x) = P_\sigma(y)\) = 0.5
\(P_\sigma(a)\) = 0.1
\(P_\sigma(b)\) = 0.4
\(P_\sigma(c)\) = 0.4
\(P_\sigma(I)\) = 0.9, where I = {a, b, c}

Player 3’s Bayesian beliefs given \(\sigma\) are

\(\mu_3(a) = \frac{P_\sigma(a)}{P_\sigma(I)} = \frac{0.1}{0.9} = \frac{1}{9}\), \(\mu_3(b) = \mu_3(c) = \frac{4}{9}\)

P(A|B) = P(A⋂B)/P(B)

When P1 plays B and P2 plays R at node y, P3’s info set I is never reached.
In this case, we say \(\mu_3\) is unrestricted and any arbitrary \(\mu_3\) is Bayesian.

Sequential Rationality

Recall that in games of perfect information, the principle of sequential rationality requires that each player’s strategy be optimal at every decision node, given the strategy profile, regardless of whether the node is reached.

Since some decision nodes are indistinguishable in games of imperfect information, we amend the principle as follows:

In games of imperfect information, the principle of sequential rationality requires that each player’s strategy be optimal at every information set, given the strategy profile and the player’s beliefs, regardless of whether the set is reached.

• Reference: Chang-Koo Chi, (36/50) Game Theory and Applications 10 – Sequential rationality in games of imperfect information, Jul 14, 2020, https://youtu.be/jEEyjlxt1ug