Terms related to Game Theory.

### Angel Problem

As proposed by J. H. Conway, a devil chases an angel on an infinite chessboard. At each move, the devil can eliminate one of the squares, and the angel can make a leap in any direction, covering a distance of at most n squares. Here, n is a positive integer previously fixed, and is called the "power" of the angel. What power angel can avoid being trapped by the devil on an island surrounded by a hole of width at least n.

### Byzantine Generals Problem

Consider a group of generals surounding a city and communicating only by messenger. The generals must agree upon a common battle plan, however, one or more of them may be traitors who will try to confuse the others.

### Conway Game

These games were introduced in 1976 to provide a formal structure for analyzing games satisfying certain requirements:

1. Two players, Left and Right, 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.

### Nim

A takeaway game played with the following rules. Given one or more piles, players alternate by taking all or some of the counters in a single heap. The player taking the last counter or stack of counters is the winner.

### Futile Game

A game which permits a draw ("tie") when played properly by both players. Tic-tac-toe, for example.

### Finite Game

A game in which each player has a finite number of moves and a finite number of choices at each move. Examples include checkers, tic-tac-toe, nim.

### Safe

A position in a game is labelled as ____ for a player, A, if the person who plays next (player B) will lose.

### Level-k Thinking

Consider a game where everyone in a classroom picks a number between 0 and 100. The person closest to half the average wins.

Obviously, picking a number over 50 would be silly. Based on this, picking a number over 25 would be silly. Similarly, picking a number over 12.5 would be silly. Continuing this line of thought, picking any number other than 0 would be silly. These thoughts show the level of thinking of those involved, from level-1 to level-inf.

### May's Theorem

This theorem states that a simple majority vote is the only procedure which is anonymous, dual, and monotonic.

### Hoyle's Social Network Theorem

A character in a novel written by an astrophysicist once opined the following. "I figure that if to be totally known and totally loved is worth 100, and to be totally unknown and totally unloved is worth 0, then to be totally known and totally unloved must be worth at least 50."

### Tit-for-Tat

A strategy for the iterated prisoner's dilemma in which a prisoner cooperates on the first move, and thereafter copies the previous move of the other prisoner. Any better strategy has more complicated rules.

### Duel

A two-person game where each of the players can take as long as he likes to prepare his move, but the other can take advantage of his hesitation. In this conflict situation, the winning strategy is entirely based on choosing the right moment for action.

### Vickrey Auction

An auction in which the highest bidder wins but pays only the second-highest bid. This variation over the normal bidding procedure is supposed to encourage bidders to bid the largest amount they are willing to pay.

### Combinatorial Game Theory

theory of two-player games of perfect knowledge such as go, chess, or checkers.

### Payoff Matrix

An m x n matrix which gives the possible outcome of a two-person zero-sum game when player A has m possible moves and player B n moves. The analysis of this matrix in order to determine optimal strategies is the aim of game theory.