# Game Theory Jacob Foley. PUfyk PUfyk

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Game Theory Jacob Foley

Overview 1. Introduction and history 2. Total-conflict games 3. Partial-conflict games 4. Three-person voting game

What is Game Theory Game- two or more individuals compete to try to control the course of events Uses mathematical tools to study situations involving both conflict and cooperation

History The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713 Theory of Games and Economic Behavior by John von Neumann in 1944 Eight game theorists have won Nobel prizes in economics

Definitions Player-maybe be people, organizations or countries Strategies- course of action they may take based on the options available to them Outcomes- the consequences of the strategies chosen by the players Preferences- each player has a perfered outcome

Game theory analyzes the rational choice of strategies Areas Applied  Bargaining tactics in labor-management disputes  Resource allocation decisions  Military Choices in international crises

What makes it different Analyzes situations in which there are at least two players The outcome depends on the choices of all the players Players can cooperate but it is not necessary

Why is it important? Provided theoretical foundations in economics Applied in political science (study of voting, elections, and international relations) Given insight into understanding the evolution of species and conditions under which animals fight each other for territory

Two-Person Total-Conflict Location Game Two Young Entrepreneurs with a new restaurant in the mountains Lisa likes low elevations Henry likes higher elevations Routes A, B, and C run east-west Highways 1, 2 and 3 run north-south Henry selects one of the routes Lisa selects one of the Highways Selection is made simultaneously

Heights of the intersections Highways Routes123 A1046 B659 C237

How do they choose Maximin- the maximum value of the minimum numbers in the row of a table Minimax- the minimum value of the maximum numbers in the columns of a table Saddlepoint- the outcome when the row minimum and the column maximum are the same

Highways Routes123Row Minima A10464 B6595 C2372 Column Maxima 1059

Solution In total-conflict games, the value is the best outcome that both players can guarantee In our example the value is 5 The value is given by each player choosing their maximin and minimax strategies

Example 2: Restricted-Location Use the same information from previous problem However, the county officials outlaw restaurants on Route B and Highway 2

Highways Routes13Row Minima A1066 C272 Column Maxima 107

Results There are no saddlepoints If both choose their minimax and maximin strategy, we will result in 7 However, they could try to out think the other which could result in 10 or 2

Duel Game Pitcher BatterFCRow Minima F.300.200 C.100.500.100 Column Maxima.300.500

Flawed Approach Pitcher- If I choose F I hold the batter down to.300 or less but the batter is likely to guess F which gives him at least.200 and actually.300 Batter- Because the pitcher will try to surprise me with C, I should guess C. I would then average.500. Pitcher- But if batter guess C, I should really throw F. Thus leading to an average of.100 for the batter

As we see…we can keep going over and over… Pure Strategy- Each of the definite courses of action that a player can choose Mixed Strategy- Course of action is randomly chosen from one of the pure strategies by:  Each pure strategy is assigned some probability, indicating the relative frequency with which that pure strategy will be played  The specific strategy used in any given play of the game can be selected at using some appropriate random device

Expected Value of E  In each of the n payoffs, s1, s2, ……, sn, will occur with the probability p1, p2, ………pn, respectively.  The expected Value E E=p1s1 +p2s2+………..+ pn*sn And we assume p1+p2+……+pn=1

Matching Pennies Two players Each has a penny They both show either heads or tails at the same time If the match, player 1 gets the pennies If they are not a match, player 2 gets the pennies

Payoff Matrix Player 2 HT Player 1H1 T 1

Results H & T are pure strategies for both players There is no way one player can outguess the other Each player should use a mixed strategy choosing H half the time and T half the time For player 1:  E(h)= ½(1) + ½(-1) = 0  E(t)= ½(-1) + ½(1) =0

Cont. The expected value for player 2 is the same This means the game is fair, which means the expected value = 0 and therefore favors neither player when at least one player uses an optimal mixed strategy If one player does not use the 50-50 strategy the player that does gains an advantage

Another example Player 2 HT Player 1H5-3 T 1

Results Player 1 E(H) = 5*(p) + (-3)(1-p) = 8p-3 E(T) = (-3)(p) +(1-p)=-4p +1  8p-3=-4p+1  12p = 4  P=1/3 Therefore,  E(H) = 8(1/3) -3 = E(T) = -4(1/3) + 1 =-1/3 => p=1/3  So their optimal mixed straigy is (1/3, 2/3) with expected value of 1/3

Cont. Using same calculations for player 2 we get the same optimal mixed stratigy of (1/3, 2/3) However, the expected value for player 2 is 1/3 Therefore, we have a zero-sum game.

Lets go back to the baseball game Pitcher BatterFCRow Minima F.300.200q C.100.5001-q Column Maxima p1-p

What should the pitcher do? E(f)= (0.3)p + (0.2)(1-p) = 0.1p + 0.2 E(c)= (0.1)p + 0.5(1-p) = -0.4p + 0.5 Solution is at the intersection of these two lines -0.4p + 0.5 = 0.1p + 0.2 p = 0.6 Giving E(f)=E(c)=E=0.26 Thus, the Pitcher should pitch F with p = 3/5 and C with p=2/5 so the batter will not be better than.260

What should the batter do? E(f)= (0.3)q + (0.1)(1-q) = 0.2q + 0.1 E(c)= (0.2)q + (0.5)(1-q) = -0.3q + 0.5 0.2q + 0.1 = -0.3q + 0.5 q=0.8 E(f) = E(c) = E = 0.260 Therefore, he should guess F with p=4/5 and C with p=1/5 which gives him a batting average of 0.260 So this gives us an outcome of 0.260

Partial-Conflict Games These are games in which the sum of payoffs to the players at different outcomes varies There can be gains by both players if the cooperate but this could be difficult

Prisoners’ Dilemma Two-person variable-sum game Shows the workings behind arms races, price wars, and some population problems In these games, each player benefits from cooperating There is no reason for them to cooperate without a credible threat of retaliation for not cooperating Albert Tucker, Princeton mathematician, named the game the Prisoners’ Dilemma in 1950

So the actual game Two people are accused of a crime Each person has a choice:  Claim their innocence  Sign a confession accusing the partner of committing the crime It is in their interest to confess and implicate their partner to receive reduce sentence However, if both confess, both will be found guilty As a team, their best interest is to deny having committed the crime

Apply it to the real world army race Two nations, Red and Blue A: Arm in preparation for war D: Disarm or negotiate an arms- control agreement Rank from best to worse (4  1) Blue AD RedA(2,2)(4,1) D(1,4)(3,3)

What should they do? Red  If Blue selects A- Red receives a payoff of 2 for A and 1 for D, so choose A  If Blue select D- Red receives a payoff of 4 for A and 3 for D, so choose A Red has a dominate strategy of A So a rational Red nation will choose A Similarly, Blue will choose A

Results If the nations work independently, we get an outcome of (A,A) with payoff of (2,2) This is a Nash Equilibrium- where no player can benefit by departing by itself from its strategy associated with an outcome So, each player can corporate, play independent, or defect Defect dominates cooperate and playing independent for both players However, defect by both players results in a worse outcome than the mutual-cooperation outcome

Another Example “Chicken” Two Drivers coming at each other at high speeds Driver 2 SwerveNot Swerve Driver 1Swerve(3,3)(2,4) Not Swerve(4,2)(1,1)

Results Neither player has a dominate strategy The Nash Equilibrium are (4,2) and (2,4) This means that getting the result of (3,3) will be unlikely because each players has an incentive to deviate to get a high payoff

Larger Games Lets look for a 3x3x3 game We find the optimal solution by looking at individuals dominant strategy Reducing it to a 3x3 game and we solve like a 2 person games we have been doing

Example: Truel A duel with 3 people Each player has a gun and can either fire or not fire at either of the other players Goal is to survive 1 st and survive with as few other players as possible http://www.youtube.com/watch?v=rExm2F bY-BE&feature=related http://www.youtube.com/watch?v=rExm2F bY-BE&feature=related

Game Tree