Brandeis

Topics in Machine Learning:
Web Projects by:

Monika Ahmetaj
mahmetaj@cs.brandeis.edu


This web page is intended to summarize and supplement Herbert Gintis' "Game Theory: A Lexicon for Strategic Interaction," and Robert Axelrod's
"Playing it Straight: Pure Strategy Nash Equilibria."

 

Game Theory: A Lexicon for Strategic Interaction

Definitions

An extensive form game G consists of:

  1. A number of players

  2. A game tree – consisting of a number of nodes connected by branches.

  3. A set of payoff

A path from a node A to a node B in the game tree is a sequence of moves starting at A and ending at B. There must be exactly one path from the root node to any given terminal node in the game tree. From this it follows that every node in the tree other than the root node must have exactly one parent.

The information set is a collection of nodes (or a node) which describes the current situation in which the player is found. The player may node the exact node on the tree when its his turn to move or he may know that he is at one of a several possible nodes.

Perfect recall is criterion that demands that if two nodes A and B belong to the same information set, then the moves made up to A and B must also be the same.

The strategic form or normal form game consists of:

  1. A set of players i = 1, … , n

  2. A set Si of Strategies for player i=1, … , n, a strategy profile for the game.

  3. A function πi: S č R for player i=1, . . . , n, where S is the set of strategy profiles, so πi(s) is the player
    i’s payoff when strategy profile s is chosen.

 

Nash Equilibrium

Suppose the game has n players, with strategy sets Si and payoff functions πi: S č R for player i=1, …………., n, where S is the set of strategy profiles.

Let sЄS and tiЄSi, we write

(t1, s2, …, sn)                                                            if i = 1
        (s-i, t-i) = (ti, s-i) =  (s1, . . . , si-1, ti, si+1, . . . , sn)         if 1 < I < n.
(s1, . . . , sn-1, tn)                                                        if i = n

We say a strategy profile s* = (si*, . . . , sn*)ЄS is a Nash Equilibrium if, for every player i=1, . . . , n, and every si ЄSi, πi(s*) ≥ πi(s-i*, si).  This means that si* is at least as beneficial for player I as choosing any other si given that other players choose s-i*.

In a Nash equilibrium, the strategy of each player is a best response to the strategies chosen by all other players.

 

Playing it Straight: Pure Strategy Nash Equilibria, by Robert Axelrod

A pure strategy Nash equilibrium of a game is a Nash equilibrium in which each player uses a pure, but not necessarily dominant, strategy.

A pure coordination game is a game in which there is one pure strategy Nash equilibriums that strictly Pareto-dominates all other Nash equilibria.

Note:  Every competitive equilibrium is Pareto-Optimal if there is no way to improve the welfare of one agent without lowering the welfare of some other agent.

A finite game is a game with a finite number of nodes in its game tree.

A game of perfect information is a game where every information-set is a single node and Nature has no moves. Every finite game of perfect information has a pure strategy Nash equilibrium.

The expected value of a lottery is the sum of the payoffs, where each payoff is weighted by the probability that the payoff will occur. There are also compound lotteries, where the payoff to one lottery is another lottery.

Let pt be a unique path from the root node to t. We say pt is compatible with strategy profile s if, for every branch
b on pt, if player i moves at bh (the head node of b), then si chooses action b at bh. 

A lottery is a set of payoffs x1, . . . , xn where xε R with probabilities p1, . . . , pn εR, where each

p1 ≥  0, and ∑1p1 = 1. The expected value of the lottery l is defined to be:

                        E[l] = ∑iεT n  pixi

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