One Shot Game, its Strategic Form and Eliminating Dominated Strategies

One Shot Game:

If a game is played only once and the players move simultaneously or at least no player knows any of the other players’ moves before choosing his. Thus we fully characterize a one-shot game by a list of the available strategies and payoffs K = {S1 ,.... , SI1,.....,πI}

1) Strategic Form:

It is called the strategic (or normal) form representation of a game. For starters, let’s consider the strategic form of a one-shot game with only two players, A and B, each with two strategies, 1 and 2. (The players could be two firms, an employer and employee and a parent and child and so on.)

The payoffs for each player are collected in the following two matrices.

556_strategic formula of two matrix.jpg

These are combined into a solo game matrix:

825_solo game matrix.jpg

Which completely summarizes the strategic form of the game. The game matrix is useful for depicting the strategic form of games with few players (usually two or three) and a finite number of strategies.

A game is symmetric if πA jk = πBkj for all j and k. If πAjk+ πBkj = c , where c is a constant, for each pair of strategies (j, k), then the game is constant sum; if c = 0, then it is a zero-sum game. Mostly in general games are variable sum.

We are seems for a solution to such games. If each player is rational, what is her optimal strategy? This is given by the best response function. Player i’s best comeback to other player’s strategies is the solution to the following maximization problem:

max πi(s1,...., Si−1, si, Si+1,....., si) ........(i)

given the strategies of the (I −1) other players. Therefore the best response function is si = Ri (si), which can also be expressed as Ri (s1,....., si−1, si+1,...., sI); that is, i’s most excellent strategy is generally a function of the strategies of all other players.

If each player plays her optimal strategy, what happens? That is, what is the equilibrium of such a game?

2) Eliminating Dominated Strategies:

One feature of best response functions is that they never reflect dominated strategies. For player i, strategy si′ dominates strategy si′′ if the payoff to si′ exceeds the payoff to si ′′ for every combination of other players’ strategies si ; that is, if π i (si ′ ,si ) > π i (si ′′ ,si )   ...........(ii)

for all si. Rational players never play dominated strategies si′′ , so we can frequently eliminate some strategies as candidates for solutions.

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