Arguments for Recovering Cooperation Conclusions that some have drawn from analysis of prisoner’s dilemma: – the game theory notion of rational action is wrong! – somehow the dilemma is being formulated wrongly – This isn’t rational. We may not defect for a few cents. If sucker’s payoff really hurts, more likely to be rational. Wool6Iterated Games1

Arguments to recover cooperation: – We are not all self-centered! But sometimes we are only nice because there is a punishment. If we don’t give up seat on bus, we receive rude stares. – If this were true, places like Honor Copy would be exploited. – The other prisoner is my twin! When I decide what to do, the other agent will do the same. (but can’t force it, as wouldn’t be autonomous). – Your mother would say, “What if everyone were to behave like that?” You say, “I would be a fool to act any other way.” – The shadow of the future…we will meet again. Wool6Iterated Games2

The Iterated Prisoner’s Dilemma One answer: play the game more than once If you know you will be meeting your opponent again, then the incentive to defect appears to evaporate Cooperation is the rational choice in the infinitely repeated prisoner’s dilemma (Hurrah!) Wool6Iterated Games3

Backwards Induction But…suppose you both know that you will play the game exactly n times On round n - 1, you have an incentive to defect, to gain that extra bit of payoff… But this makes round n – 2 the last “real”, and so you have an incentive to defect there, too. This is the backwards induction problem. Playing the prisoner’s dilemma with a fixed, finite, pre- determined, commonly known number of rounds, defection is the best strategy Wool6Iterated Games4

The centipede game – What would you do? Either player can stop the game. Wool6Iterated Games5 Jack stop (2, 0) Go on Jill stop (1, 4) Jill Go on stop (5, 3) Jack Go on stop (4, 7) Jill (98, 96) stop (99, 99) Go on (97, 100) stop Go on Jack Go on Jill (94, 97)

The centipede game Wool6Iterated Games6 Jack stop (2, 0) Go on Jill stop (1, 4) Jill Go on stop (5, 3) Jack Go on stop (4, 7) Jill (98, 96) stop (99, 99) Go on (97, 100) stop Go on Jack Go on Jill (94, 97) The solution to this game through backward induction is for Jack to stop in the first round!

The centipede game What actually happens? In experiments the game usually continues for at least a few rounds and occasionally goes all the way to the end. But going all the way to the (99, 99) payoff almost never happens – at some stage of the game ‘cooperation’ breaks down. So still do not get sustained cooperation even if move away from ‘roll back’ as a solution Wool6Iterated Games7

Lessons from finite repeated games – Finite repetition often does not help players to reach better solutions – Often the outcome of the finitely repeated game is simply the one-shot Nash equilibrium repeated again and again. – There are SOME repeated games where finite repetition can create new equilibrium outcomes. But these games tend to have special properties – For a large number of repetitions, there are some games where the Nash equilibrium logic breaks down in practice. Wool6Iterated Games8

Axelrod’s Tournament Suppose you play iterated prisoner’s dilemma against a range of opponents… What strategy should you choose, so as to maximize your overall payoff? Axelrod (1984) investigated this problem, with a computer tournament for programs playing the prisoner’s dilemma Wool6Iterated Games9

Axelrod’s tournament: invited political scientists, psychologists, economists, game theoreticians to play iterated prisoners dilemma All-D – always defect Random: randomly pick a strategy Tit-for-Tat – On first round cooperate. Then do whatever your opponent did last. Tester – first defect, If the opponent ever retaliates, then use tit-for-tat. If the opponent does not defect, cooperate for two rounds, then defect. Joss: Tit-for-tat, but 10% of the time, defect instead of cooperating. Which do you think had the highest scores? Wool6Iterated Games10

Best? Tit-for-Tat Why? Because you were averaging over all types of strategy If you played only All-D, tit-for-tat would lose. Wool6Iterated Games11

Two Trigger Strategies Grim trigger strategy – Cooperate until a rival deviates – Once a deviation occurs, play non-cooperatively for the rest of the game Tit-for-tat – Cooperate if your rival cooperated in the most recent period – Cheat if your rival cheated in the most recent period Wool6Iterated Games12

Axelrod's rules for success Do not be envious – not necessary to beat your opponent in order to do well. This is not zero sum. Do not be the first to defect. Be nice. Start by cooperating. Retaliate appropriately: Always punish defection immediately, but use “measured” force — don’t overdo it Don’t hold grudges: Always reciprocate cooperation immediately do not be too clever – when you try to learn from the other agent, don’t forget he is trying to learn from you. – Be forgiving – one defect doesn’t mean you can never cooperate – The opponent may be acting randomly Wool6Iterated Games13

Threats Threatening retaliatory actions may help gain cooperation Threat needs to be believable – “If you are late for class, I will give you an F” - credible? – “If you break the rules, you will be grounded for a year.” – credible? – “If you cross me, we will not go trick-or-treating.” – credible? Wool6Iterated Games14

What is Credibility? “The difference between genius and stupidity is that genius has its limits.” – Albert Einstein You are not credible if you propose to take suboptimal actions.: A rational actor proposes to play a strategy which earns suboptimal profit. How can one be credible? Wool6Iterated Games15

Trigger Strategy Extremes Tit-for-Tat is – most forgiving – shortest memory – proportional – credible but lacks deterrence Tit-for-tat answers: “Is cooperation easy?” Grim trigger is – least forgiving – longest memory – MAD – adequate deterrence but lacks credibility Grim trigger answers: “Is cooperation possible?” Wool6Iterated Games16

concepts of rationality [doing the rational thing] undominated strategy (problem: too weak) can’t always find a single undominated strategy (weakly) dominating strategy (alias “duh?”) (problem: too strong, rarely exists) Nash equilibrium (or double best response) (problem: equilibrium may not exist) randomized (mixed) Nash equilibrium – players choose various options based on some random number (assigned via a probability) Theorem [Nash 1952]: randomized Nash Equilibrium always exists. Wool6Iterated Games

Mixed strategy equilibria  i (s j ) is the probability player i selects strategy sj (0,0,…1,0,…0) is a pure strategy (over n possible choices) Strategy profile:  =(  1,…,  n ) Expected utility: chance the outcome occurs times utility Nash Equilibrium: –  * is a (mixed) Nash equilibrium if Wool6Iterated Games18  i  i defines a probability distribution over Si ui(  * i,  * -i )  ui(  i,  * -i ) for all  i  i, for all i

Example: Matching Pennies no pure strategy Nash Equilibrium Wool6Iterated Games19 -1, 11,-1 -1, 1 H HT T Pure strategy equilibria [I make one choice.]. Not all games have pure strategy equilibria. Some equilibria are mixed strategy equilibria.

Example: Matching Pennies Wool6Iterated Games20 -1, 11,-1 -1, 1 p H q H1-q T 1-p T Want to play each strategy with a certain probability. If player 2 is optimally mixing strategies, player 1 is indifferent between his own choices! Compute expected utility given each pure possibility of other player.

I reason about my choices as player 2 21 Note, my concern is in how well the other person is doing because I know he will be motivated to do what is best for himself If I pick q=1/2, what is my strategy?

I am player 2. What should I do? I pick a defensive strategy If player1 picks heads, his opponent gets: -q+(1-q) If Player 1 picks tails, his opponent gets: q + -(1-q) Want my opponent NOT to care what I pick. The idea is, if my opponent gets excited about what my strategy is, it means I have left open an opportunity for him. When it doesn’t matter what he does, it says there is no way he wins big. So: -q +(1-q) =q + -1+q 1-2q=2q-1 so q=1/2 Wool6Iterated Games22

Example: Bach/Stravinsky Wool6Iterated Games23 2, 10,0 1, 2 p B q B1-q S 1-p S Want to play each strategy with a certain probability. If player 2 is optimally mixing strategies, player 1 is indifferent to what player1 does. Compute expected utility given each pure possibility of yours.

Example: Bach/Stravinsky Wool6Iterated Games24 2, 10,0 1, 2 p B q B1-q S 1-p S Want to play each strategy with a certain probability. If player 2 is optimally mixing strategies, player 1 is indifferent to what player1 does. Compute expected utility given each pure possibility of yours. p = 2(1-p) p=2/3 2q = (1-q)q=1/3 player 1 is optimally mixing player 2 is optimally mixing

Mixed Strategies Unreasonable predictors of one-time human interaction Reasonable predictors of long-term proportions Wool6Iterated Games25

Employee Monitoring Employees can work hard or shirk Salary: $100K unless caught shirking Cost of effort: $50K (We are assuming that when he works he loses something. Think of him as having to pay for resources to do his job – expensive paper, subcontracting, etc. We are also assuming that unless the employee is caught shirking the boss can’t tell he hasn’t been working.) Managers can monitor or not Value of employee output: $200K (We assume he must be worth more than we pay him to cover profit, infrastructure, manager time, mistakes, etc.) Profit if employee doesn’t work: $0 Cost of monitoring: $10K Give me the normal form game payoffs Wool6Iterated Games26

Employee Monitoring From the problem statement, VERIFY the numbers in the table are correct. No equilibrium in pure strategies - SHOW IT What do the players do in mixed strategies? DO AT SEATS Please do not consider this instruction for how to cheat your boss. Rather, think of it as advice in how to deal with employees. 27 Manager (q) Monitor (q) No Monitor (1-q) Employee Work (p) 50, 9050, 100 Shirk (1-p) 0, , -100

p – probability of working q – probability of monitoring 28

Employee’s Payoff First, find employee’s expected payoff from each pure strategy If employee works: receives 50 Profit(work)= 50  q+ 50  (1-q) = 50 If employee shirks: receives 0 or 100 Profit(shirk)= 0  q+ 100  (1-q) = 100 – 100q Wool6Iterated Games29

Employee’s Best Response Next, calculate the best strategy for possible strategies of the opponent For q<1/2: SHIRK Profit (shirk) = q > 50 = Profit(work) SHIRK For q>1/2: WORK Profit (shirk) = q < 50 = Profit(work) WORK For q=1/2: INDIFFERENT Profit(shirk) = q = 50 = Profit(work) ???? Wool6Iterated Games30

Cycles 31 q 01 1/2 p 0 9/10 1 work shirk monitorno monitor If I am not monitoring and they are working, they will change their mind

Properties of Equilibrium Both players are indifferent between any mixture over their strategies E.g. employee: If shirk: If work: Regardless of what employee does, expected payoff is the same Similar for employer. Their utility is 80 Wool6Iterated Games32

Upsetting? This example is upsetting as it appears to tell you, as workers, to shirk. Think of it from the manager’s point of view, assuming you have unmotivated (or unhappy) workers. A better option would be to hire dedicated workers, but if you have people who are trying to cheat you, this gives a reasonable response. Sometimes you are dealing with individuals who just want to beat the system. In that case, you need to play their game. For example, people who try to beat the IRS. On the positive side, even if you have dishonest workers, if you get too paranoid about monitoring their work, you lose! This theory tells you to lighten up! Wool6Iterated Games33

Why Do We Mix? I don’t want to give my opponent an advantage. When my opponent can’t decide what to do based on my strategy, I win – as there is not way he is going to take advantage of me. Wool6Iterated Games34 COMMANDMENT Use the mixed strategy that keeps your opponent guessing.

The following example is one you can work through on your own. 35

Mixed Strategy Equilibriums Anyone for tennis? – Should you serve to the forehand or the backhand? Wool6Iterated Games36

Tennis Payoffs Wool6Iterated Games37

Tennis: Fixed Sum If you win (the points), I lose (the points) AKA: Strictly competitive Wool6Iterated Games38 q1-q p 1-p

Solving for Server’s Optimal Mix What would happen if the the server always served to the forehand? – A rational receiver would always anticipate forehand and 90% of the serves would be successfully returned. Wool6Iterated Games39

Solving for Server’s Optimal Mix What would happen if the the server aimed to the forehand 50% of the time and the backhand 50% of the time and the receiver always guessed forehand? – (0.5*0.9) + (0.5*0.2) = 0.55 successful returns Wool6Iterated Games40

Solving for Server’s Optimal Mix What is the best mix for each player? Receiver thinks: if server serves forehand.10*p +.70*(1-p) if server serves backhand.80*p +.40*(1-p) I want them to be the same.10*p +.70*(1-p) =.80*p +.40*(1-p).10*p p =.80*p p -.6p+.7 =.4p =p Use similar argument to solve for q – So strategies are ((.3,.7)(.4,.6)) Wool6Iterated Games41

Draw a graph which shows two lines (1) the utility of server of “picking forehand” as a function of p. (2) the utility of server of “picking backhand” as a function of p. Wool6Iterated Games42

Receiver’s view depending on opponent action Above 1/3, backhand wins. Wool6Iterated Games43 p

Server’s view dependent on opponent action Above.4 plan forehand wins Wool6Iterated Games 44 q

% of Successful Returns Given Server and Receiver Actions Wool6Iterated Games45 Where would you shoot knowing the other player will respond to your choices? In other words, you pick the row but will likely get the smaller value in a row.