Beliefs and mixed strategies
Game Theory // Fall 2025
Prof. Santetti
marcio.santetti@emerson.edu
Beliefs
Beliefs
Recall:
- Strategy: A complete contingent plan of action.
A plan of action is conditional on what the other player(s) will do.
- So how does a player think about the other player(s)’s actions?
Beliefs
A belief is a player’s assessment about the strategies of the other player(s) in the game.
Beliefs & penalty kicks
Beliefs
How to formalize beliefs?
Beliefs → Likelihoods → Probabilities
Beliefs
Kane’s strategy space: \(S_1 = \{\text{Left}, \text{Right}\}\) (discarding Center).
Goalkeeper’s beliefs → Likelihoods → Probabilities:
- p: Likelihood that Kane will shoot to the right;
- (1 - p): Likelihood that Kane will shoot to the left.
- What if p = 1/2?
- p and (1 - p): A probability distribution over the strategy set \(\{\text{Left}, \text{Right}\}\).
Why bother?
Why bother?
People do randomize.
If we do randomize, we adopt mixed strategies.
A mixed strategy for a player is the act of selecting a strategy according to a probability distribution.
A penalty-kick game
A penalty-kick game
A penalty-kick game
