Situations where participants can all gain or suffer together are referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with.
A | B | C | |
1 | |||
2 |
A game's payoff matrix is a convenient representation. Consider for example the two-player zero-sum game pictured at right.
The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.
Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.
Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?
Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. But what happens if Blue anticipates Red's reasoning and choice of action 1, and goes for action B, so as to win 10 points? Or if Red in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?
Émile Borel and John von Neumann had the fundamental and surprising insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute provably optimal strategies for all two-player zero-sum games.
For the example given above, it turns out that Red should choose action 1 with probability 4/7 and action 2 with probability 3/7, while Blue should assign the probabilities 0, 4/7, and 3/7 to the three actions A, B, and C. Red will then win 20/7 points on average per game.
:Minimize: ::: :Subject to the constraints: :: ≥ 0 :: ≥ 1.
The first constraint says each element of the vector must be nonnegative, and the second constraint says each element of the vector must be at least 1. For the resulting vector, the inverse of the sum of its elements is the value of the game. Multiplying by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.
If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.
The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of (adding a constant so it's positive), then solving the resulting game.
If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general.
See also:
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Category:Non-cooperative games Category:International relations theory
ar:مجموع صفري ca:Joc de suma nul·la cs:Hra s nulovým součtem de:Nullsummenspiel es:Juego de suma cero fr:Jeu à somme nulle ko:제로섬 게임 is:Núllsummuleikur it:Gioco a somma zero he:משחק סכום אפס nl:Nulsomspel ja:ゼロ和 no:Nullsumspill nds:Nullsummenspeel pl:Gra o sumie stałej pt:Soma-zero ru:Антагонистическая игра sk:Hra s nulovým súčtom sr:Нулта сума fi:Nollasummapeli sv:Nollsummespel th:เกมศูนย์ uk:Ігри антагоністичні ur:صفر حاصل جمع zh:零和博弈This text is licensed under the Creative Commons CC-BY-SA License. This text was originally published on Wikipedia and was developed by the Wikipedia community.
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