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Linear Programming
Published in Albert G. Holzman, Mathematical Programming, 2020
As an introduction to game theory, consider the game of matching pennies. In the game are two players, designated as P1 and P2. Each player chooses either heads or tails. After the choice is made by each player without the knowledge of the other player, their choices are made known. If the pennies match, Player P1 wins, and P2 pays P1 one penny. If the pennies do not match, P2 wins, and P1 pays P2 one penny. A payoff matrix is set up in game theory to show the results of the strategies of the competing players. The payoff matrix for the penny matching game is shown in Fig. 3. This matrix shows a +1 payment when the pennies match. Thus, if both players choose heads or tails, P2 pays P1 one penny. The matrix also shows a -1 payment when the pennies do not match, which indicates a payment from P1 to P2. The plus and minus signs indicate the direction of the payment, where the plus payment shows payment to P1 from P2, and the minus payment indicates a payment to P2 from P1.
Game Theory
Published in Vikrant Sharma, Vinod Kumar Jain, Atul Kumar, An Introduction to Optimization Techniques, 2021
Vikrant Sharma, Vinod Kumar Jain, Atul Kumar
In general, a two-person game is characterized by the following:A two-person game includes each player’s strategies and the payoff matrix.The payoff matrix displays the benefit for Player 1 (positive or negative) that would result from each combination of the two players’ strategies. Note that the matrix for Player 2 is the negative of the matrix for Player 1 in a zero-sum game.The payoff matrix entries may be in any unit, provided they represent the player’s utility (or value).
Game Theory
Published in N.V.S. Raju, Operations Research, 2019
Payoff Matrix: A two-person zero-sum game is conveniently represented by a matrix as shown below. The matrix, which shows the outcome of the game as the players select their particular strategies, is known as the payoff matrix. It is important to assume that each player knows not only his own list of possible courses of action but also of his opponent.
Evolutionary game analysis of pedestrian-autonomous vehicle interactions at unsignalized road sections: a policy intervention perspective
Published in Transportation Letters, 2022
Rong Rui, Xusheng Yao, Shunqiang Ye, Shoufeng Ma
Players, strategies, and payoff functions are three basic elements in the strategic form of traditional game theory (Fudenberg and Tirole 1991). Players make decisions; in this paper, they are pedestrians, AVs, and traffic managers. Strategies refer to a player’s choices in terms of certain issues, e.g., price or effort. Usually, the profits or utilities of a player in a game are called payoffs. The payoff matrix can succinctly represent the strategies and corresponding payoffs of different players.