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Randy Whitehead

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What is a Game? We all know how to play games. Whether they involve cards, sports equipment, boards, dice, or a multitude of other things, but is that the extent of what we can call a game? A game is defined as an activity engaged in for activity or amusement. With this in mind, can we really restrict a game to an arbitrary set of equipment used? The reality of the situation is that every interpersonal reaction can be reduced to a game with different strategies, outcomes, and payoffs.

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Representation of Games There are two main types used to represent a game in the field of game theory.

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Extensive This method is often used in games where players alternate moves and you can look back and see all that has happened before, often called a game of perfect information, but it can also be used for simultaneous games as well.

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Extensive Example This example shows the very first move made in a chess game. As you can see, it can get very large very quickly.

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Strategic This method is favored by most people because it sets up all possible choices and outcomes into a single payoff matrix. Game theorists will often try to reduce a game into one or a collection of 2 x 2 payoff matrices.

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Strategic Example This example shows a simple number calling game between two players. The payoffs are written with the corresponding number equivalent to the amount that player 1 pays player 2, i.e. if the number is negative then player 2 pays player 1 the absolute value of the number.

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General Strategic Form Let player R have m possible moves and player C have n possible moves. Depending on the two players' moves, a payoff is made from player C to player R.

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General Strategic Form For all i = 1,2,...,m, and j = 1,2,...,n, let us set a ij = payoff that player C makes to player R if player R makes move i and player C makes move j Therefore, the payoff matrix is:

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General Strategic Form The next natural step is to examine the probability that a player will make a certain move p i =probability that player R makes move i. q j =probability that player C makes move j.

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General Strategic Form With the probabilities p i and q j, two vectors can be formed: Where the row vector p is called the strategy of player R and the column vector q is called the strategy of player C.

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Theorem 1 For a 2 x 2 game that is not strictly determined, optimal strategies for players R and C are: The value of the game is:

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Inoculation Strategies The federal government desires to inoculate its citizens against a certain flu virus. The virus has two strains, and the proportions in which the two strains occur in the virus population is not known. Two vaccines have been developed. Vaccine 1 is 85% effective against strain 1 and 70% effective against strain 2. Vaccine 2 is 60% effective against strain 1 and 90% effective against strain 2. What inoculation policy should the government adopt?

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Inoculation Strategies We may consider this a two-person game in which player R (the government) desires to make the payoff (the fraction of citizens resistant to the virus) as large as possible, and player C (the virus) desires to make the payoff as small as possible. The payoff matrix is:

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Inoculation Strategies This matrix is not strictly determined because it has no saddle points. Therefore, Theorem 1 is applicable.

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Prisoner’s Dilemma Suppose the police have captured two suspects that they know have committed armed robbery together but do not have enough evidence to convict either of the suspects of the armed robbery charge, but they do have evidence of the suspects stealing the getaway car. The chief officer then makes the following offer to each prisoner: "If you will confess to the robbery, implicating your partner, and she does not also confess, then you'll go free and she'll get ten years. If you both confess, you'll each get 5 years. If neither of you confess, then you'll each get two years for the auto theft."

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Prisoner’s Dilemma We can model the situation with a payoff matrix and assign cardinal numbers to each outcome years in prison 2 5 years in prison 3 2 years in prison 4 0 years in prison

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Prisoner’s Dilemma By using math, which isn’t shown in this presentation, we arrive at the optimal strategy for the prisoner’s dilemma. According to game theory, the prisoner should defect in all situations regardless of what the other prisoner does.

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Prisoner’s Dilemma If this example is applied to more relatable applications, the results become much more disturbing. Let’s take a look at the Cold War for example.

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Cold War The Cold War between the United States and the Soviet Union can be modeled directly as a prisoner’s dilemma with the following payoff matrix:

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Cold War According to game theory, a rational player would increase their amount of arms in all situations. Luckily, the United States took the more peaceful approach and the Soviet Union followed suit. If both entities had continued to increase their amount of arms, the world would be a much different and more dangerous place.

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