[Update] Cypher Substance Game

 ZERO SUM GAME


In game theory and economic theory, a zero-sum game is a mathematical representation of a province of affairs inwards which each participant's gain or loss of utility is just balanced yesteryear the losses or gains of the utility of the other participants. If the full gains of the participants are added upward in addition to the full losses are subtracted, they volition total to zero. Thus, cutting a cake, where taking a larger slice reduces the amount of cake available for others, is a zero-sum game if all participants value each unit of measurement of cake as (see marginal utility).

In contrast, non-zero-sum describes a province of affairs inwards which the interacting parties' aggregate gains in addition to losses tin flame endure less than or to a greater extent than than zero. Influenza A virus subtype H5N1 zero-sum game is besides called a strictly competitive game spell non-zero-sum games tin flame endure either competitive or non-competitive. Zero-sum games are virtually oft solved amongst the minimax theorem which is closely related to linear programming duality,^[1] or amongst Nash equilibrium.

Example

A zero-sum game

Blue Red	A	B	C
1	-30 30	10 -10	-20 20
2	10 -10	-20 20	20 -20

Influenza A virus subtype H5N1 game's payoff matrix is a convenient representation. Consider for event the two-player zero-sum game pictured at correct or above.
The gild of play proceeds as follows: The starting fourth dimension thespian (red) chooses inwards hole-and-corner i of the 2 actions 1 or 2; the minute thespian (blue), unaware of the starting fourth dimension player's choice, chooses inwards hole-and-corner i of the 3 actions A, B or C. Then, the choices are revealed in addition to each player's points full is affected according to the wages for those choices.
Example: Red chooses activeness 2 in addition to Blue chooses activeness B. When the wages is allocated, Red gains xx points in addition to Blue loses xx points.
In this event game, both players know the wages matrix in addition to endeavour to maximize the release of their points. Red could argue as follows: "With activeness 2, I could lose upward to xx points in addition to tin flame win solely 20, in addition to amongst activeness 1 I tin flame lose solely 10 but tin flame win upward to 30, hence activeness 1 looks a lot better." With like reasoning, Blue would select activeness C. If both players withdraw maintain these actions, Red volition win xx points. If Blue anticipates Red's reasoning in addition to alternative of activeness 1, Blue may select activeness B, hence as to win 10 points. If Red, inwards turn, anticipates this push clitoris a fast i on in addition to goes for activeness 2, this wins Red xx points.
Émile Borel in addition to John von Neumann had the telephone commutation insight that probability provides a agency out of this conundrum. Instead of deciding on a definite activeness to take, the 2 players assign probabilities to their respective actions, in addition to and hence role a random device which, according to these probabilities, chooses an activeness for them. Each thespian computes the probabilities hence as to minimize the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming work amongst the optimal strategies for each player. This minimax method tin flame compute in all likelihood optimal strategies for all two-player zero-sum games.
For the event given above, it turns out that Red should select activeness 1 amongst probability 4/7 in addition to activeness 2 amongst probability 3/7, in addition to Blue should assign the probabilities 0, 4/7, in addition to 3/7 to the 3 actions A, B, in addition to C. Red volition in addition to hence win 20/7 points on average per game.

Solving

The Nash equilibrium for a two-player, zero-sum game tin flame endure flora yesteryear solving a linear programming problem. Suppose a zero-sum game has a wages matrix M where chemical ingredient ${\displaystyle M_{i,j}}$ is the wages obtained when the minimizing thespian chooses pure strategy i in addition to the maximizing thespian chooses pure strategy j (i.e. the thespian trying to minimize the wages chooses the row in addition to the thespian trying to maximize the wages chooses the column). Assume every chemical ingredient of M is positive. The game volition withdraw maintain at to the lowest degree i Nash equilibrium. The Nash equilibrium tin flame endure flora (see ref. [2], page 740) yesteryear solving the next linear computer program to detect a vector u:

Minimize:

${\displaystyle \sum _{i}u_{i}}$

Subject to the constraints:

u ≥ 0

M u ≥ 1.

The starting fourth dimension constraint says each chemical ingredient of the u vector must endure nonnegative, in addition to the minute constraint says each chemical ingredient of the M u vector must endure at to the lowest degree 1. For the resulting u vector, the inverse of the total of its elements is the value of the game. Multiplying u yesteryear that value gives a probability vector, giving the probability that the maximizing thespian volition select each of the possible pure strategies.
If the game matrix does non withdraw maintain all positive elements, merely add together a constant to every chemical ingredient that is large plenty to brand them all positive. That volition increase the value of the game yesteryear that constant, in addition to volition withdraw maintain no resultant on the equilibrium mixed strategies for the equilibrium.
The equilibrium mixed strategy for the minimizing thespian tin flame endure flora yesteryear solving the dual of the given linear program. Or, it tin flame endure flora yesteryear using the inwards a higher house physical care for to solve a modified wages matrix which is the transpose in addition to negation of M (adding a constant hence it's positive), in addition to hence solving the resulting game.
If all the solutions to the linear computer program are found, they volition constitute all the Nash equilibria for the game. Conversely, whatsoever linear computer program tin flame endure converted into a two-player, zero-sum game yesteryear using a alter of variables that puts it inwards the shape of the inwards a higher house equations. So such games are equivalent to linear programs^.

                                                           nix total game .