Game Theory
Game theory looks at the actions of rational decision makers. A decision maker’s strategy is determined after considering all possible situations that incorporate the interests of an opponent. Game theory does not insist on how a game should be played but rather provides the procedure by which actions should be selected. Thus, it is a decision theory useful in competitive situations.
A game represents a competitive or conflicting situation between two or more players. Each player has a number of choices that are called moves. A strategy for a given player is a plan that specifies which of the available choices should be made. It is important to explicitly assume that the players are rational, where they act to maximize their expected payoffs (or minimize their expected losses).
With a game theory model, we provide a mathematical description of a social situation in which two or more players interact. Consider a competitive situation with two players, Player I and Player II. The game is specified by the sets of strategies available to the two players and the payoff matrix. The set of strategies for Player I is indexed from 1 through m while the set of strategies for Player II is indexed from 1 through n. The payoff matrix (Table 1) specifies the gain or profit to Player I for every strategy pair (i, j). The two players select their strategies simultaneously, and when Player I chooses strategy i and Player II uses strategy j, Player I receives the payoff pij from Player II. A positive number represents a gain for Player I while a negative number signifies a loss (a gain for Player II). One player’s winnings are the other player’s losses so the net gain is zero across all players, providing the zero-sum feature. The payoff obtained when the two players select their strategies is the value of the game.
Table 1: Payoff matrix for a two-person, zero-sum game.
Game theory is applied to determine the optimal strategies for each player. In light of the conflicting nature and lack of information between the strategies selected by each player, the optimal strategy guarantees a payoff that can never be worsened by the selections of the opponent. This criterion is known as the “minimax criterion” (or “maximin criterion”). In general terms, the pure minimax strategy for the row player is the strategy that maximizes his minimum gain. The payoff for this strategy is denoted by the value of the lower bound VL:
VL = Max{row minimum} = Maxi = 1,…,m{Minj = 1,…,n pij}
The pure minimax strategy for the column player is the strategy that minimizes the maximum gain for the row player, setting an upper bound on the payout designated by VU:
VU = Min(column maximum) = Minj = 1,…,n{Maxi = 1,…,m pij}
The value (V) of the game is when the quantities VL and VU defined above are the same. When the upper and lower values are equal the game has a saddle point. A saddle point is an equilibrium point in that neither player can benefit from an alternative strategy. An easy way to spot a saddle point is to observe that the reward for a saddle point must be the smallest number in its row and the largest number in its column. A pair of strategies (k, l) is said to be a saddle point if pil pkl pkj for all i and j. In any zero-sum game, the maximin minimax. If the game has a saddle point (k, l) then maximin = minimax = pkl. Additionally, if a game has a saddle point, then the maximin and minimax coincide with that of a Nash equilibrium. A Nash equilibrium is a solution where neither player can change his strategy to improve his payout under the assumption that each player knows the equilibrium strategies of the other players. If each player has chosen a strategy where no player can independently improve his benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitutes a Nash equilibrium.
Let’s look at an example of a two-person zero-sum game that involves a row player and column player, where the payoff matrix represents what the row player receives from the column player.
Table 2: A game with a saddle point.
The row player wants to maximize his payout. If he chooses row 1, the column player will choose columns 1 or 2 to only pay four units rather than ten units. Similarly, if the row player decides on row 2 then the column player will decide on column 3 to reduce the payout to one unit. Therefore, the row player will select a row that has the largest minimum, which in this case is row 3, ensuring that he will win at least the row minimum of five units. The greatest payoff the row player can guarantee himself is the maximin.
From the column player’s viewpoint, he wants to minimize his losses. If the column player selects column 1, then the row player will choose the strategy that makes the column player’s losses as large as possible (and the row player’s winnings as large as possible). Thus, if the column player chooses column 1, then the row player will select row 3, to maximize his payout with six. Similarly, if the column player chooses column 2, then the row player will again choose row 3 to maximize his payout. Lastly, selecting column 3 will lead to a loss of ten units because the row player is looking to maximize his payout. Consequently, the column player can minimize his losses by selecting column 2. The lowest payoff the column player can ensure the row player receives is the minimax.
Overall, the row player can ensure that he will win at least five units and the column player can hold the row player’s winnings to at most five units. Thus, the only rational outcome of this game is for the row player to win exactly five units. When the row player decides on another row, the column player can respond with a strategy to diminish the payout to the row player.
The two players are using the minimax criterion for strategy selection. The row player will follow the strategy Max{4, 1, 5} = 5. The row that determines the maximum is the pure minimax strategy for the row player, which for this example is row 3. The column player will abide by the Min{6, 5, 10} = 5. The column that obtains the minimum is the pure minimax strategy for the column player, which in this case is column 2. In this game, both players can adopt the pure minimax strategy and cannot improve their positions by moving to any other strategy; therefore, (3, 2) is the saddle point of the game.
Before looking at another example, it is important to carefully review an assumption of game theory: the players are rational decision makers. Players should aim to make themselves as best off as possible; however, that is not always the case. The first game to look at is the ultimatum game. The first player, the proposer, receives a sum of money, for argument’s sake say $100, and proposes how to divide the total between oneself and the other player. The second player, the responder, chooses to either accept or reject the first player’s proposal. If the responder rejects the offer, both players get nothing. If the proposer decides to keep $90 for himself and offer $10 to the second player, the responder might decide to reject the offer. Humans are economically irrational in that they take into account factors other than direct economic payoffs. Most people find more personal utility in enforcing social equity than walking away with a monetary benefit. As a result, player two rejects the offer and walks away with nothing. The second game is the dictator game. The first player is given a sum of money, let’s use $100 again, and has to decide how much to keep for himself and how much to give to the second player. Note that this is not really a game but rather a look at decision theory since the second player has no say in the final outcome. This game provides evidence against the notion that humans are rationally self-interested. Humans are concerned with self-image and altruism so they, in most cases, would not decide to keep the total $100 to themselves but rather provide the second player with a portion of the sum.
Understanding that humans are not always rational is important as we look at another game called golden balls. The way the game works is each player has to either split or steal without the knowledge of the other player’s decision.
Table 3: Golden balls game.
The first number in each cell represents the percentage of the pot Person 1 is entitled to and the second number represents the percentage of the pot Person 2 receives. For Person 1, the better option is to steal if Person 2 splits to raise his percentage of the pot from 50 percent to 100 percent. On the other hand, Person 1 would be indifferent between splitting and stealing if Person 2 decides to steal. Therefore, this leaves the steal option to be weakly dominant. Similarly, Person 2 is indifferent when Person 1 steals and would prefer to steal if Person 1 splits, so the option to steal is weakly dominant for Person 2. This would result in cell (2, 2) being a Nash equilibrium as there is no move that would make one of the players better off given that they are both aware of the other stealing. Interestingly, cell (1, 2) is also a Nash equilibrium since Person 1 would not be better off switching from split to steal given that Person 2 is stealing. The same logic can be applied for cell (2, 1), leaving Table 3 with a total of three Nash equilibriums. Note that cell (1, 1) is not a Nash equilibrium. If Person 2 decides to split, then Person 1 would deviate from splitting and prefer to steal and leave with 100 percent of the pot rather than just 50 percent. Overall, the incentives of both players are to steal.
What if Person 2 understands game theory? Before both players elect to split or steal, Person 2 tells Person 1 that he is automatically going to steal, thereby eliminating the left hand side of the payoff matrix. Consequently, Person 1 would be left with 0 percent of the pot regardless of splitting or stealing, leaving him indifferent. Further, Person 2 says to trust him and that he would be willing to share half of the pot afterward if Person 1 decides to split rather than steal. This leaves Person 1 leaning to split simply because there is the upside that Person 2—if trustworthy—would share half the pot after the game.
When both players then decide to split or steal, both players reveal that they have elected to split. Person 1 decided to choose the rational, self-interest play that includes the potential of receiving half the pot by electing to split rather than steal. Person 2 decided to split for an important reason that reflects his understanding of how humans can act irrationally. It is possible that Person 1 had no faith in Person 2 and would prefer to spite Person 2 and gain personal utility by not letting anyone have access to the pot. Therefore, Person 2 elects to split so both players do not leave empty-handed in the case that Person 1 still decides to steal.
The Simplex Method can be used to determine the optimal solution for a particular game. The row player always attempts to choose the set of strategies with the non-zero probabilities, say p1, p2, p3 where p1 + p2 + p3 = 1 maximizes his minimum expected gain (maximin).
The objective of the row player is to maximize the value V, which is equivalent to minimizing the value 1/V. The linear program is written as the following:
Min 1/V = p1/V + p2/V + p3/V
and constraints ≥ 1.
This is equivalent to the following:
Min1/V = X1 + X2 + X3
and constraints ≥ 1.
Similarly, the column player would choose the set of strategies with the non-zero probabilities, say q1, q2, q3 where q1 + q2 + q3 = 1 minimizes his maximum expected loss (minimax).
This yields the dual of the above linear program that acts as the primal.
Max 1/V = q1/V + q2/V + q3/V
and constraints ≤ 1.
This is again equivalent to the following:
Max 1/V = Y1 + Y2 + Y3
and constraints ≤ 1.
The dual provides a maximization problem that can be solved using the Simplex Method. Each row in the payoff matrix represents a constraint that should be no greater than one. Reverting back to Table 2, we can use the Simplex Method to determine the optimal solution for the two-person, zero-sum game. The three constraints encode the three rows found in the payoff matrix.
Figure 1: The formulas used to encode the payoff matrix from Table 2.
After encoding the payoff matrix, the optimal solution is found to be 1/5. This represents the maximum to the dual. Therefore, if the objective function, which maximizes 1/V, can be bounded at 1/5, then V is equal to 5. This is in accordance with the saddle point that was determined earlier.
Figure 2: The optimal solution to the linear program that encodes Table 2.
Prisoner’s dilemma
punishment payoff: both confess and screw each other (8,8)
whoever gets the 10 is the sucker payoff…you try to cooperate and your teammate does not
whoever gets the 0 is the temptation payoff because you did confess (received best payoff possible)
made yourself best off as possible
cooperation payoff: both get -2 and don't confess
8,8 is a Nash equilibrium because the reactionary responses to make yourself best off depending on the other player’s course of action
If only one iteration
2,2 is weak Nash eq (only if it is weakly iterated)…you can improve yourself but if you improve from 2 to 0 years and he has the same option and it would put you at 8,8
Tit for tat with multiple iterations
Game example: Ultimatum game
People take into account things other than direct economic payoffs!
More personal utility from social equity than monetary benefits
People take into account things other than money
Walk in with $0 and someone offers you $$$ (pot of 100) and you can accept/reject the offer, say the person offers 70 for himself and 30 for you…ppl would reject the 30 because they perceive it as unfair
They punish the inequity and avoid getting a free $30
Gets personal utility so you do not get your $70
Game: Dictator game
By show of hands, how many of you would take the full $100???
Social aspect…people won’t raise their hands
= Altruism and greed and social aspect
Proves altruism
I can choose how much I get of the $100 and how much the person across from me gets.
People do not want to come across with negative social perception.
Usually people give about half to each.
Altruism…people gain utility by seeing others gain
stag hunt game theory
Two Nash equilibriums…(coop,coop) (defect, defect)
No enticing/temptation payoff for me to cooperate if he is defecting unless we both switch to cooperate
Mixed equilibrium – probability that makes me indifferent between two decisions based upon the other person cooperating
battle of the sexes
two pure nash equilbirums – the corners with nonzero
never switching to (0,0)
Game theory was invented by John von Neumann and Oskar Morgenstern in 1944. Over the past 60 years, their framework has been expanded, deepened, and made more general to apply to several situations. While it was invented in 1944, game theory theoretical insights can be dated back to ancient times. In Plato’s texts, Laches and Symposium, a situation involving a soldier at the front waiting with his comrades to fight off an enemy attack utilizes the game theory framework. The problem the soldier faces is whether he should stay and fight or run for safety. If he thinks his side will win, then he could essentially flee and it would not matter. Otherwise, he could stay and possibly die for no reason. If he thinks his side will lose, he could run for safety and live or he could stay and die. As a result, the soldier is always better off running away. Along these same lines, military generals have been using this thinking for years. This conflict is the earliest forms that we have seen of game theory.
In the 1940s, the concept of game theory expanded beyond just a way of reasoning to its present form that supports the decision making process. Thomas Hobbes, a founder of modern political philosophy, pushed this idea of game theory as a form of logic in his book, Leviathan. Hobbes talks about the idea of house building in that if you help build my house with the promise of me helping to build your house, what keeps me to the promise after I get what I want? Hobbes solution to these problems was the tyranny. The idea of punishment after a broken promise would keep people honest. These issues are game theory from the beginning.
John von Neumann and Oskar Morgenstern’s book Theory of Games and Economic Behaviors, changed the idea of Economics as it was able to be applied to so many other concepts. It was able to be applied to war, politics, recreational games, and psychology. The theory was created using many models that had paved the way to the creation of Game Theory including Brouwer’s fixed-point theorem. From his book and the field of Game Theory, the classic model of the Prisoner’s dilemma was created. The two put into words, the dilemmas and situations that philosophers and war generals had been facing for years.
Maggie “final” draft:
Game theory was invented by John von Neumann and Oskar Morgenstern in 1944. Over the past 60 years, their framework has been expanded, deepened, and made more general to apply to several situations. While it was invented in 1944, Game Theory theoretical insights can be dated back to ancient times. In Plato’s text, the Laches and the Symposium, a situation involving a soldier at front waiting with his comrades to fight off an enemy attack. The problem that the soldier faces is whether he should stay and fight or run for safety. If he thinks his side will win, then he could essentially flee and it would not matter. Or he could stay and possibly die for no reason. If he thinks he side will lose, he could run for safety and live or he could die by staying. The solution that Plato discusses is theoretically the solider is always better off running away. Along these same lines, military generals have been using this thinking for years. This conflict is the earliest forms that we have seen of Game Theory
However, this logic was simply logic until the 1940’s when math was able to be put behind the decision making process. Hobbes pushed this idea of game theory as a form of logic before his time. He explored these ideas in his book, Levithan. Hobbes talks about the idea of house building in that if you help build my house with the promise of me helping to build your house, what keeps me to the promise after I get what I want? Hobbes solution to these problems was the tyranny. The idea of punishment after a broken promise would keep people honest. These issues are game theory from the beginning.
John von Neumann and Oskar Morgenstern’s book Theory of Games and Economic Behaviors, changed the idea of Economics as it was able to be applied to so many other concepts. It was able to be applied to war, politics, recreational games, and psychology. The theory was created using many models that had paved the way to the creation of Game Theory including Brouwer’s fixed-point theorem. From his book and the field of Game Theory, the classic model of the Prisoner’s dilemma was created. The two put into words, the dilemmas and situations that philosophers and war generals had been facing for years.