 # Game theory

## Game theory Topics

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### Wythoff's game

A game played with two heaps of counters in which a player may take any number from either heap or the same number from both. The player taking the last counter wins. The th safe combination is , where , with the golden ratio and the floor function. It is also true that . The first few safe combinations are (1, 2), (3, 5), (4, 7), (6, 10), ... (OEIS A000201 and A001950), which are the pairs of elements from the complementary Beatty sequences for and (Wells 1986, p. 40).

### Equilibrium point

An equilibrium point in game theory is a set of strategies such that the th payoff function is larger or equal for any other th strategy, i.e.,

### Nash equilibrium

A Nash equilibrium of a strategic game is a profile of strategies , where ( is the strategy set of player ), such that for each player , , , where and .Another way to state the Nash equilibrium condition is that solves for each . In words, in a Nash equilibrium, no player has an incentive to deviate from the strategy chosen, since no player can choose a better strategy given the choices of the other players.The Season 1 episode "Dirty Bomb" (2005) of the television crime drama NUMB3RS mentions Nash equilibrium.

### Monty hall problem

The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do.The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat. But after Monty has eliminated one of the doors for you, you obviously do not improve..

### Early election results

Let Jones and Smith be the only two contestants in an election that will end in a deadlock when all votes for Jones () and Smith () are counted. What is the expectation value of after of a total of votes are counted? The solution is (1)(2)

### Minimax theorem

The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. It was proved by John von Neumann in 1928.Formally, let and be mixed strategies for players A and B. Let be the payoff matrix. Thenwhere is called the value of the game and and are called the solutions. It also turns out that if there is more than one optimal mixed strategy, there are infinitely many.In the Season 4 opening episode "Trust Metric" (2007) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that he used the minimax theorem in an attempt to derive an equation describing friendship.

### Domineering

A two-player game, also called crosscram, in which player has horizontal dominoes and player has vertical dominoes. The two players alternately place a domino on a board until the other cannot move, in which case the player having made the last move wins (Gardner 1974, Lachmann et al. 2000). Depending on the dimensions of the board, the winner will be , , 1 (the player making the first move), or 2 (the player making the second move). For example, the board is a win for the first player.Berlekamp (1988) solved the general problem for board for odd . Solutions for the board are summarized in the following table, with a win for for .winwinwin0210120H1V11121H2112H22H311322314H14124H5V15125H6116H26H7117H2718H18128H9V19129HLachmann et al. (2000) have solved the game for widths of , 3, 4, 5, 7, 9, and 11, obtaining the results summarized in the following table for , 1, ....winner32, V, 1, 1, H, H, ...4H for even and all 52, V, H, V, H, 2, H, H, ...7H for 9H for..

### Social choice theory

The theory of analyzing a decision between a collection of alternatives made by a collection of voters with separate opinions. Any choice for the entire group should reflect the desires of the individual voters to the extent possible.Fair choice procedures usually satisfy anonymity (invariance under permutation of voters), duality (each alternative receives equal weight for a single vote), and monotonicity (a change favorable for does not hurt ). Simple majority vote is anonymous, dual, and monotone. May's theorem states a stronger result.

### Conway game

Conway games were introduced by J. H. Conway in 1976 to provide a formal structure for analyzing games satisfying certain requirements: 1. There are two players, Left and Right ( and ), who move alternately. 2. The first player unable to move loses. 3. Both players have complete information about the state of the game. 4. There is no element of chance. For example, nim is a Conway game, but chess is not (due to the possibility of draws and stalemate). Note that Conway's "game of life" is (somewhat confusingly) not a Conway game.A Conway game is either: 1. The zero game, denoted as 0 or , or 2. An object (an ordered pair) of the form , where and are sets of Conway games. The elements of and are called the Left and Right options respectively, and are the moves available to Left and Right. For example, in the game , if it is 's move, he may move to or , whereas if it is 's move, he has no options and loses immediately.A game in which both players have..

### Sharing problem

A problem also known as the points problem or unfinished game. Consider a tournament involving players playing the same game repetitively. Each game has a single winner, and denote the number of games won by player at some juncture . The games are independent, and the probability of the th player winning a game is . The tournament is specified to continue until one player has won games. If the tournament is discontinued before any player has won games so that for , ..., , how should the prize money be shared in order to distribute it proportionally to the players' chances of winning?For player , call the number of games left to win the "quota." For two players, let and be the probabilities of winning a single game, and and be the number of games needed for each player to win the tournament. Then the stakes should be divided in the ratio , where(1)(2)(Kraitchik 1942).If players have equal probability of winning ("cell probability"),..

### Quota system

A generalization of simple majority voting in which a list of quotas specifies, according to the number of votes, how many votes an alternative needs to win (Taylor 1995). The quota system declares a tie unless for some , there are exactly tie votes in the profile and one of the alternatives has at least votes, in which case the alternative is the choice.Let be the number of quota systems for voters and the number of quota systems for which , so(1)where is the floor function. This produces the sequence of central binomial coefficients 1, 2, 3, 6, 10, 20, 35, 70, 126, ... (OEIS A001405). It may be defined recursively by and(2)where is a Catalan number (Young et al. 1995). The function satisfies(3)for (Young et al. 1995). satisfies the quota rule.

### Game value

The solution to a game in gametheory. When a game saddle point is presentand is the value for pure strategies.

### Prisoner's dilemma

A problem in game theory first discussed by A. Tucker. Suppose each of two prisoners and , who are not allowed to communicate with each other, is offered to be set free if he implicates the other. If neither implicates the other, both will receive the usual sentence. However, if the prisoners implicate each other, then both are presumed guilty and granted harsh sentences.A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision. Each prisoner's best strategy would appear to be to turn the other in (since if makes the worst-case assumption that will turn him in, then will walk free and will be stuck in jail if he remains silent). However, if the prisoners turn each other in, they obtain the worst possible outcome for both.Mosteller (1987) describes a different problem he terms "the prisoner's dilemma." In this problem, three prisoners , , and with apparently equally good records..

### Game theory

Game theory is a branch of mathematics that deals with the analysis of games (i.e., situations involving parties with conflicting interests). In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real-world problems as diverse as economics, property division, politics, and warfare.Game theory has two distinct branches: combinatorialgame theory and classical game theory.Combinatorial game theory covers two-player games of perfect knowledge such as go, chess, or checkers. Notably, combinatorial games have no chance element, and players take turns.In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.The..

For a general two-player zero-sum game,If the two are equal, then writewhere is called the value of the game. In this case, there exist optimal strategies for the first and second players.A necessary and sufficient condition for a saddle point to exist is the presence of a payoff matrix element which is both a minimum of its row and a maximum of its column. A game may have more than one saddle point, but all must have the same value.

### Payoff matrix

An matrix which gives the possible outcome of a two-person zero-sum game when player A has possible moves and player B moves. The analysis of the matrix in order to determine optimal strategies is the aim of game theory. The so-called "augmented" payoff matrix is defined as follows:

### Game expectation

Let the elements in a payoff matrix be denoted , where the s are player A's strategies and the s are player B's strategies. Player A can get at least(1)for strategy . Player B can force player A to get no more than for a strategy . The best strategy for player A is therefore(2)and the best strategy for player B is(3)In general,(4)Equality holds only if a game saddle point is present, in which case the quantity is called the value of the game.