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Finite mathematics topic summary: game theory 
TwoPerson Zero Sum Game
In a twoperson zero sum game, each of the two players is given a choice between several prescribed moves at each turn, and each player's loss is equal to the other player's gain. The payoff matrix of a twoperson zero sum game has rows labeled by the "row player's" moves and columns labeled by the opposing "column player's" moves. The ij entry of the matrix is the payoff that accrues to the row player in the event that the row player uses move i and the column player uses move j. 
Example
Paper, Scissors, Rock
Do you want to play? Click on a row strategy...


Mixed Strategy, Expected Value
A player uses a pure strategy if he or she uses the same move at each round of the game. A mixed strategy is a method of playing a game where, at each round of the game, the player chooses a move at random so that each is used a predetermined fraction of the time. We represent a mixed (or pure) strategy for the row player by a row matrix (probability vector) R = [a b c . . . ]with the same number of entries as there are rows, where each entry represents the fraction of times the corresponding row is played (or the probability of using that strategy) and where a + b + . . . = 1. A mixed strategy for the column player is represented by a similar column matrix C. For both row and column players, pure strategies is represented by vectors in with a single 1 and the rest zeros. The expected value of the game with payoff matrix P corresponding to the mixed strategies R and C is given by e = RPCThe expected value of the game is the average payoff per round if each player uses their mixed strategy for a large number of rounds. 
Example
Here is a variant of "paper, scissors, rock in which "paper/paper" and "rock/rock" is no longer a draw.
Suppose the row player uses the mixed strategy R = [0.75 0 0.25](play paper 75% of the time, scissors 0% of the time and rock 25% of the time) and the column player plays scissors and rock each 50% of the time; Then the expected value of the game is


Minimax Criterion, Fundamental Principles of Game Theory
Minimax Criterion
Finding the minimax strategy is called solving the game. In the third part of the tutorial for this topic we show a graphical method for solving 2×2 games. For general games, one uses the simplex method (see the next topic summary). However, one can frequently simplify a game and sometimes solve it by "reducing by dominance" and/or checking whether it is "strictly determined" (see below). Fundamental Principles of Game Theory When analyzing any game, we make the following assumptions about both players:

Example
Consider the following game.
If the row player follows Principle 1, (s)he should never play Move 1 because Move 2 gives better payoffs no matter what move the column player chooses. (The payoffs in Row 2 are all at least as high as the corresponding ones in Row 1.) Further, following Principle 2, the row player expects that the column player will never play Move A, since Move B gives better payoffs as far as the column player is concerned. (The payoffs in Column B are all at least as low as the corresponding ones in Column A.) 

Reducing by Dominance
One move dominates another if all its payoffs are at least as advantageous to the player than the corresponding ones in the other move. In terms of the payoff matrix, we can say it this way:
Following the first principle of game theory, the move corresponding to a strictly dominated row or column will never be played, and both players are aware of this by the second principle. Thus each player following the principles of game theory will repeatedly eliminate dominated rows and columns as the case may be. (If two rows or columns are equal, then there is no reason to choose one over the other, so either may be eliminated.) This process is called reduction by dominance. 
Example
Consider the above game once again.
Since the entries in Row 2 are ≥ the corresponding ones in Row 1, Row 2 dominates Row 1. Since the entries in Column B are ≤ the corresponding ones in Column A, Column B dominates Column A. Reducing the Above Game by Dominance Since Row 2 dominates Row 1 we eliminate Row 1 to obtain
Since Column B now dominates both Columns A and C we eliminate both Columns A and C to obtain
Since the top row now dominates the bottom row, we eliminate the bottom row, and we are reduced to the following 1×1 matrix
In this case, we have solved the game by reduction by dominance: The row player should always play 2 and the column player should always play B. Since the corresponding payoff is 0, we say that the game is fair (neither player has an advantage over the other). Note that we were lucky here: Not all games can be reduced by dominance to a 1×1 game. 

Saddle Point, Strictly Determined Game
A saddle point is a payoff that is simultaneously a row minimum and a column maximum. To locate saddle points, circle the row minima and box the column maxima. The saddle points are those entries that are both circled and boxed. A game is strictly determined if it has at least one saddle point. The following statements are true about strictly determined games.
The value of a strictly determined game is the value of the saddle point entry. A fair game has value of zero, otherwise it is unfair or biased. 
Example
In the above game, there are two saddle points, shown in color.
Since the saddle point entries are zero, this is a fair game. The online game theory utility can be used to check any game (up to 5×5) for saddle points. Try it out. 