Cybersecurity and Applied Mathematics by Unknown

Author:Unknown
Language: eng
Format: epub

6.2 The mathematical definition of a game 97

Ta b le 6 . 2 The Payoff Matrix for the

Prisoner’s Dilemma

a pair. Does she choose what is best for herself or does she choose what is best for the group.

Rather than using the terms silent and confess , this game sometimes uses the terms cooperate and defect to make it clear that it is about the group versus the individual. The player that stays silent cooperates with the group for the greater good of the group. The player that confesses defects from the common good for the betterment of the individual.

6.2 THE MATHEMATICAL DEFINITION OF A GAME

In the previous section we discussed the prisoner’s dilemma with two players. There is actually no rule that requires that game to be played with only two players, that is just the simplest version of the game. In this section we will discuss ≥ the mathematical definition of a game with n players. We will assume that n 2 as we are not discussing single player games. Game theory is a study about competition. This means that there must be at least two players for the competition to exist.

6.2.1 STRATEGIES, PAYOFFS AND NORMAL FORM

The set of players is { 1, 2, ... ,∩ n } .=∅ .Each player i has a set of strategies (or moves) S i. It is entirely possible that S i S j The set of moves = may nor may not be disjoint. From the prisoner’s dilemma game, we know that S 1 S 2 . If we consider the game between the network defender and the network attacker, then we do not expect them to be using the same strategies.

Each move has a potential payoff. So if there are m strategies for a player, there are ∈ m payoffs. The set ∈of payoffs for each player’s strategies is U i. So each strategy s S i has a payoff u U i. Each payoff is a real number and is associated with a strategy, so rather than writing this as a separate set we can write it as a function. It is not just the player’s strategy that can affect the payoff, but the other players in the games can affect the payoff. × We ×··· Ssaw this → in the prisoner’s dilemma. The payoff function is a function u i : S 1 S 2 m R .

98 CHAPTER 6 Game theory

The easiest × case → to visualize is the two player game. u 1 : S 1 × S 2 → R

and u 2 : S 1 S 2 R are the payoff functions. We can list all of the possible

payoff outcomes in matrix form. We will let A 1 be the payoff matrix for player 1. Each element = of the matrix a i,j is the value of the function u 1 at ( s i, s j) .In short, a i,j u 1 ( s i, s j) . In a similar fashion we can create the matrix A 2 for player 2.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.