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