Theory of Games

1. Introduction: five first lessons

Defining the game: outcome matrix and payoffs

outcome matrix: common representation of the outcomes of actions of the different players is a table, like this:

The second part of defining a game is the objective/goals. We map the outcomes to payoffs, which are numerical values where higher is better, and lower is worse.

A game is composed of:

• players
• set of possible strategies for each player. A strategy profile is a tuple of strategies, one for each player.
• Payoffs.

Basic properties of strategies

Dominant strategy: Domination of a strategy over another strategy if the strategy's payoff is better than the other regardless of the other's player strategy.

His example is the Prisioner's dilemma, where "rational choice" may not be optimal. If both agreed, then they could do better. Repeated interaction could allow cooperation. We not only need more than communication, we need enforcement, a Contract... See Law. Also Education. All this helps by changing the payoffs. Video about all this

Variation of the game with guilt/altruism, and indignation, gives rise to a Coordination problem.

Combination of the two games –> taking into account what the other player will do Important in the number game he made –> gives rise to a sequence of i know you know, etc, called Common knowledge in philosophy. Assuming common knowledge of rationality, the optimal choice would have been 1.

Algorithmic game theory

Two-player finite zero-sum games

$\max\min u(x,y) \leq \min \max u(x,y)$

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