## Beliefs and Expected Utility

People will choose the strategy that maximizes their payoff the most.

Game theory assumes that all players are Bayesian rational.

Bayesian rationality requires players to form **beliefs** about how opponents will act

What this means First, each player has prior beliefs about all the events involved in the payoff.

All information is complete, the only uncertainty is not knowing what the other player will do.

Each player’s actions affect the other player’s payoff.

Consider a two-player game

- Suppose that P1 expects P2 to play a pure strategy, but P1 is uncertain of which strategy P2 will play
- Then player 1 should form a belief \(\theta_{1}\ \in \Delta S_{2}\) about what player 2 will do.

Specifically, for each \(s_{2} \in S_{2}\).

\(\theta_{1}(s_{2})\) = player 1’s assessment that player 2 chooses \(s_{2}\), and \(\sum_{s_{i} \in S_{i}}\theta_{1}(s_{2})\) = 1

theta is a function, s is a pure strategy

e.g., \(S_{2}\) = {L, C, R} – with probabilities 30, 60, and 10, respectively

Then, one of the beliefs the player can have, \(\theta_{1} = \frac{3}{10}L + \frac{6}{10}C + \frac{1}{10}R\)

\(\theta_{1}(2)\) = 30%

So if you add everything up, you get 1

- Note that such a belief takes the same form of a mixed strategy of player 2
- Player 1’s expected payoff given \(\theta_{1}\) is

\(u_{1}(s_{1},\theta_{1}) = \sum_{s_{i} \in S_{i}}\theta_{2}u_{1}(s_{1}, s_{2})\)

(\Delta S_{2}) is the sum of all pure and mixed strategies that player 2 can choose.

Player 1 believes that the probability of player 2 choosing \(s_{2}\) is \(\theta_{1}(s_{2})\).

In reality, if player 2 uses the \(s_{2}\) strategy, player 1’s payoff will be \(u_{1}(s_{1}, s_{2})\).

The probability of occurrence multiplied by the actual realized payoff and summed over all possible \(S_{2}\) is the expected payoff.

EXAMPLE 2.4.

L | M | R | |

U | 8, 1 | 0, 2 | 4, 0 |

C | 3, 3 | 1, 2 | 0, 0 |

D | 5, 0 | 2, 3 | 8, 1 |

Suppose \(\theta_{1}\) = (1/4, 1/2, 1/4).

\(u_{1}(U, \theta_{1}) = u_{1}(U, L)\theta_{1}(L) + u_{1}(U, M)\theta_{1}(M) + u_{1}(U, R)\theta_{1}(R)\)

= 8 ∙ \(\frac{1}{4}\) + 0 ∙ \(\frac{1}{2}\) + 4 ∙ \(\frac{1}{4}\) = 3.

The sum of all of these is the expected payoff.

Similary, the expected payoff from playing a mixed strategy \(\sigma_{1}\) = (1/2, 0, 1/2) is

\(u_{1}(\sigma_{1}, \theta_{1}) = u_{1}(U, \theta_{1})\sigma_{1}(U) + u_{1}(C, \theta_{1})\sigma_{1}(C) + u_{1}(D, \theta_{1})\sigma_{1}(D)\)

= 3 ∙ \(\frac{1}{2}\) + \(\frac{5}{4}\) ∙ 0 + \(\frac{17}{4}\) ∙ \(\frac{1}{2}\) = \(\frac{29}{8}\)

∵ \(u_{1}(D, \theta_{1}) = 5 ∙ \frac{1}{4} + 2 ∙ \frac{1}{2} + 8 ∙ \frac{1}{4} = \frac{(5+4+8)}{4} = \frac{17}{4}\)

Given a belief \(\theta_{i}\), a Bayesian-rational player will choose a strategy \(s_{i}\) that maximizes her expected utility, put another way, player 1 would choose the optimal strategy,

\(\sigma_{1}^* \in argmax_{\sigma_{1} \in \Delta S_{1}} u_{1}(\sigma_{1}, \theta_{1})\)

Player 1’s optimal strategy is which strategy (sigma) maximizes player 1’s expected payoff given player 1’s beliefs (theta).

Bayesian-rational requires that the prediction is that player 1 will eventually choose this.

In the previous example, pure strategy D yields the highest expected utility given \(\theta_{1}\).

Therefore, if player 1 is a Bayesian rational DM with \(\theta_{1}\), she would choose D.

#### Summary

The first of the three elements of a normal-form game, the player has little to say about it.

The second element, strategy, has pure and mixed strategies, of which the mixed strategy is based on independence.

Using the payoff function, a Bayesian-rational player can say which strategy to choose, depends on the beliefs about how the opponent will move.

We want to calculate the expected payoff for a given belief and choose the strategy that maximizes the expected payoff.

- Reference: Chang-Koo Chi, (4/50) Game Theory and Applications 2 – Normal-form games, basic concepts, Jul 1, 2020, 05:36-22:47, https://youtu.be/YzD3DoYagYg