Lecture 3: Prisoner’s Dilemma, Tit-for-tat, Ultimatum and Dictator's Games
1. Labor law is all about PD . Taft-Hartley requires unions to represent all workers, including non-members. Some states allow agency fees, where non-members pay for services; RTW laws forbid such fees. Indiana which does not allow fees has some depts in small cities where all workers vote for union but then refuse to join and pay – the ultimate “free rider”/defect in the PD. June 27, 2018 Supreme Court issues 5-4 Janus decision that government workers cannot be forced to pay union fees, endangering the economic status of public sector unions. The Court rejected the amica curie from “Petitioner’s assumptions contradict decades, if not centuries, of economic theory and empirical evidence. As Mancur Olson demonstrated, a rational employee motivated solely by economic self-interest will withhold union dues or fair-share fees if he c an do so without incurring countervailing costs— even if he benefits from the union, believes he benefits, and agrees with the union’s actions on his behalf— because his fees “alone would not perceptibly strengthen the union, and since he would get the benefits of any union achievements whether or not he supported the union.” Mancur Olson, The Logic of Collective Action. https://www-cdn.law.stanford.edu/wp-content/uploads/2018/02/Brief-of-Amici-Curiae-Economists-and-Professors- of-Law-and-Economics-in-Support-of-Respondents.18Jan2018.pdf.pdf Did this end state and local govt protests and unions? Check out teacher strikes in 2018 organized outside of unions 2.Environmental policy and tragedy of commons issues are prisoners'-dilemma (PD) games. A country that participates in reducing emissions bears the full cost of the reduction, but gains part of the climate benefit. A country that reduces oil consumption bears the full cost of the reduction, but gains only a small part of the savings from lower oil prices.
Ostrom found that societies offer many different solutions to tragedy of commons problem depending on nature of resource (and governance, etc). Many related games with same basic structure/different numeric values in matrix.
The Prisoner’s Dilemma (PD): Two players cooperate or defect. If A cooperates and B defects, A loses; if A defects and B cooperates A wins; if both cooperate both gain; if both defect, both lose. Standard PD payoff matrix
The payoffs: CC> DC/CD > DD; and CC>[ DC + CD]/2 (so alternating strategy is not best). Since CC produces max: 6 > 5> 2, PD is not a zero sum game. Cooperation has a productive value. CC can be socially negative -- cartel ripping off consumers. Often written as T(DC) > R (CC) > P (DD) > S(CD) with R= 3, S=0, T=5, P =1 In a one-shot game, solution is D. If I/you do C, I/you will do D and win. So we both play D. Same holds if we know the game ends in T.,which is a one-shot game. But T-1 is a one-shot game again. And so on. Thus a known number of interactions yields ALLD. But much less defecting in the world. Why? 1. Expected future dealing, producing Rip off the tourist but cooperate with your spouse. If expect to interact again with no certain endpoint, get repeated or iterated PD game: IPD, for which there is no best action. Returns depend on what others do and discount of future payoffs. 2. Low discount rate (= 1/(1+r)) so future dealings matter. If we do CC with r=0 we each get 3 ad finitum compared to a possible 5 and 1's if both defect thereafter. 3+3+3 + … > 5+ 1+1 + … Future matters more when r is small so strategy that pays off in future can beat various defect strategies. 3. Conditional retaliatory strategies. If I play D against your C and you do not change to D, I win. If you shift to D against me, I get 5+1 in two rounds and 5+1+1 in three rounds while if you play C with another C player you get 6 in two rounds and 9 in three rounds and 9>7. Retaliation drops D to 1 in next rounds. Key other strategy is TFT, tit-for-tat, cooperate until opponent defect, then defect until opponent changes. An eye for an eye, tooth for tooth. Opponent C D C/D TFT C C D C/D
4.World of many strategies beyond all D or C In all-D world, best is D. In all C world, best is D. But with other strategies may be better to be nicer. Three periods is minimum for TFT to work better than All D given the payoff matrix above where TFT and D meet half the time. For simplicity let r=0 TFT meets TFT: rewards = 3+3+3) =9 TFT meets All D: rewards = 0 +1 +1 = 2 All D meets TFT: rewards = 5 +1 +1 =7 All D meets All D: rewards = 1+ 1+1 = 3 TFT gets 11 (= 9+2 ) from playing D and TFT; D gets 10 (=7+3). TFT cooperation > defect.
5.Winning strategy varies with the distribution of strategies in world. In all-D world best is all-D. In TFT world, best is TFT type strategy. Consider how payoffs vary with the all-D and TFT population in a 3 period model % D TFT D 1/3 20/3 (1/3 2 + 2/3 9) 17/3 (1/3 3 + 2/3 7) TFT WINS 1/2 11/2 (½ 2 +1/2 9) 10/2 TFT WINS 2/3 13/3 13/3 EQUAL SCORES 3/4 15/4 16/4 D WINS So when %D> 2/3rds, D wins; when %D < 2/3rds D loses; at 2/3rds get unstable mixed equilibrium. Note TFT requires smaller proportion of itself to win (1/3rd +) than D (2/3rd+). Reason is 6>5.
6. Addition of all Cooperate (turn other cheek) helps all-D and hurts TFT: Too many suckers destroys world
TFT C D 1/3 of each 2/5 TFT, 2/5C, 7/10 TFT 2/10 C TFT 9 9 2 20/3 38/5 8.3* *FOR WIN C 9 9 0 18/3 36/5 8.1 D 7 15 3 25/3* 47/5* 8.2
D wins because it exploits C. With 2/5 TFT and 2/5 C (and 1/5 D), D wins. With 7/10 TFT and 2/10 C, TFT wins.
Thus, NO BEST CHOICE IN iterated PD. SUCCESS DEPENDS ON ECOLOGY OF STRATEGIES. For any payoff matrix, there is a distribution of All D, All C, and TFT so that D wins and that TFT wins. One on one, C never wins. TFT never wins as D always scores more. The key to cooperation is that nice strategies gain from interactions with nice strategies. TFT beats D through its interaction with TFT. PD game on TV http://gawker.com/5903692/must-watch-golden-balls-contestant-wins-with-most-ballsy-move-ever
Axelrod 1979 Computer Tournament R. Axelrod asked experts to submit programs for the PD – code giving responses to any action by another. Fifteen programs enter, including D and C. Several complex programs try to infer and exploit opponents strategy. Anatol Rapaport enters TFT. TFT wins. Axelrod announces results and holds second contest. Analysis of round 1 showed that a more generous/ forgiving strategy could beat TFT: Tit for two tats -- TFTT -- which retaliates against DD but not D. 63 entrants in 2nd tournament and TFT (Rapaport) won again. Axelrod then simulated what would happen to the population of strategies in the next generation if higher scoring strategies increase their share of the population – evolutionary process. TFT and other nice rules did well over time. Why? TFT/nice strategies win because they never defect first but retaliate quickly to D, which limits D's points. Can a TFT world survive invasion of Ds? Depends on %D invades (p). In first period TFT scores 3(1-p) + p, while D gets (1-p) + 5p so TFT beats D when 2(1-p)> -4p >0 ---> p<2/3. So if population change depends on relative scores, initial invasion of <2/3Ds would fail. Conversely, a world of Ds can survive invasion of TFTs but p< 1/3. So easier for TFT to defend against invasion or more needed with given matrix. But note that TFT world cannot repel invasion of Cs? because TFT and C score the same. Cs open door to D invasion. Spatial interactions and n-hoods If TFTs interact more with each other in local N-HOOD rather than with the entire population TFT is more likely to survive. Say 1-p% TFTs enter All-D and have 2 of their 4 interactions with TFTs. Then their score is equivalent to a world with 50% TFTs. But the Ds still interact largely with Ds, so TFT could win. Cellular Automata PD models show how n-hood interactions and the “lattice” affect outcomes in spatial PD games. Assume that players interact with others in nhood and change strategy depending on what wins in the Nhood. Surrounded by Ds you turn D. Surrounded by TFTs you go TFT. Conflicts occur on the borders. Compare a TFT with 3 Ds and 1 TFT for neighbors with a TFT and D having half TFT neighbors and a TFT with 2 TFT neighbors. TFT TFT TFT TFT * ?? * TFT D * ?? * D D * ?? * D D TF T D ?? computes profits from D and TFT and picks most profitable. Consider the rewards using payoffs for three period interactions: TFT-TFT 9, TFT-D 2, D-D 3, D-TFT 7
NEIGHBORHOOD PICK 1D, 3 TFT 2D 2 TFT 3 D 1 TFT TFT 29 22 15 Surrounded by 2 or 3 TFTs choose TFT. D 24 20 16 Surrounded by 3 or more Ds choose D; Decision TFT TFT D Go to http://ccl.northwestern.edu/netlogo/models/PDBasicEvolutionaryl and experiment with the PD games. Additional stuff on spatial interactions 1)Review of experiments on Prisoner’s Dilemmas on lattices suggest that imposed lattice structure does not influence global cooperation, (Grujik, et al, 2014 A comparative analysis of spatial Prisoner’s Dilemma experiments:Conditional cooperation and payoff irrelevance, www.nature.com/articles/srep04615 2)Simulation of a zealot who cooperates irrespective of the result of an interaction destroys rather than boosts cooperation. (Matsuzawa,et al “Spatial prisoner’s dilemma games with zealous cooperators” PHYSICAL REVIEW E 94, 022114 (2016) Better than TFT: Nicer and Conditional TFT has problems with errors in communication D'. If TFT meets TFT and errs, it --> an alternating cycle, with lower rewards than C. TFT CCC D' CDCD …More forgiving is TFTT CCC D' CC CCC DD CC TFT CCC C DCDC... TFTT CCC C CC. CCC CC DD To generalize strategies via conditional probabilities, let P be the probability you cooperate if X cooperated and Q be the probability you cooperate if X defected. This gives strategies below (Sigmund, Games of Life,)
P: You Cooperate if Other Person cooperated last period Nowak&Sigmund simulate world of (p,q) strategies with random ps and qs and NO neighborhoods. PAVLOV responds to previous round by switching if it loses: if its D leads to a D, it tries C ; if its C meets D, it tries D. WIN-STAY; LOSE-SHIFT. Pavlov would fail in Axelrod-tournament until TFT destroys most Ds..
Psychology Experiments-- Framing matters( if these are replicable) Study 1: More cooperation in ‘‘Community Game’’ PD than ‘‘Wall Street Game’’ in Israeli Air force. Instructors guessed who will cooperate based on behavior during training. (Liberman, V., S. M. Samuels, and L. Ross. 2004. Personality and Social Psychology Bulletin 30:1175-85.) Study 2: Interpretive labels of the game, the choices, and the outcomes led to different outcomes. (Zhong , Loewenstein, Murnighan “ Journal of Conflict Resolution,” Vol. 51, No. 3, 431-456 (2007))
Study 3: Can people predict who will cooperate in PD (NB Israeli instructors did not predict well)
ULTIMATUM GAME If A moves first, what is rational for A to get B to say “OK”? 1 cent, and you are better off. If B moves first, what is rational for B to do to get A to say “OK, I accept this deal” – demand all but 1 cent.
Meta-analysis of 97 ultimatum games in 42 articles between 1983 and 2012. Weighted average offer by the proposers is 41%. Country of experiment and being an economist affect amounts offered. (http://metaanalysis2014.econ.uoa.gr/fileadmin/metaanalysis2014.econ.uoa.gr/uploads/Tisserand_Jean- Christian.pdf
Social Psych Framing – Acceptances are significantly influenced by offer fairness and type of description Framing in terms of features of the other player impact decisions. Two-hundred and forty undergraduates played the UG after being provided with different descriptions of the Proposer's (no information, physical description, psychological description). These results support the relevance of the expectation effects due to the framing in social decision making. (Marchetti, Castelli, Harle and Sanfey Expectations and outcome: The role of Proposer features in the Ultimatum Game Journal of Economic Psychology, 2011, vol. 32, issue 3, 446-449) Game presented on a printed booklet and participants provided their responses directly on it. Participants played Responder, receiving a one-time monetary offer from a Proposer, with four levels of description: 1. No information (control condition). 2. Physical description: ‘‘Mary/Peter is 20 years old. She/he is tall with brown eyes and dark hair. She/he usually dresses in a serious fashion’’. 3. Brief positive psychological description (generous condition): ‘‘Mary/Peter is a generous and altruistic person. She/he always takes into consideration the point of view of other people, as well as their needs. Mary/Peter thinks that respect and fairness are fundamental values in human relationships’’. 4. Brief negative psychological description (selfish condition): ‘‘Mary/Peter is a selfish and suspicious person. She/he first takes into consideration her/his own goals and own interests. Mary/Peter thinks that ensuring one’s own success is a fundamental value in human relationships’’. Participants received the offer with respect to a division of 100 Euros (approx $150). Participants were offered either a ’fair’ proposal of 40 Euros, or an ‘unfair’ proposal of 10 Euros. Eight groups in total were tested, comprised of two sets of offers (fair, unfair) crossed with four levels of partner description (none, physical, generous, selfish) in a between-subjects design.
Notice: Best rip-off deal is from generous described people. Why? No analysis of gender/income/etc. Study talked only about Proposer of same gender. What is going on inside people's head ... NEURO Hypothesis :emotional part of brain overcomes rational part in ultimatum game/other decisions
The neural basis of economic decision-making in the Ultimatum Game. Sanfey et al Science. 2003 Jun 13;300(5626):1755-8. used functional magnetic resonance imaging of Ultimatum Game players to investigate neural substrates of cognitive and emotional processes involved in economic decision-making...scanned players as they responded to fair and unfair proposals. Unfair offers elicited activity in brain areas related to both emotion (anterior insula) and cognition (dorsolateral prefrontal cortex). Further, significantly heightened activity in anterior insula for rejected unfair offers suggests an important role for emotions in decision-making.
Irrational Economic Decision-Making after Ventromedial Prefrontal Damage: Evidence from Ultimatum Game M.Koenigs and D Tranel: “Relatively low Ultimatum offers are often rejected, and this “irrational” behavior has been attributed to an emotional reaction to unfair treatment...we tested hypothesis that damage to ventromedial prefrontal cortex (VMPC), an area critical for the modulation of emotional reactions, would result in exaggerated irrational economic decisions. Subjects acted as responder to 22 different proposers who offered various splits of $10. Offers ranged from fair (give $5, keep $5) to extremely unfair (give $1, keep $9). The rejection rate of the VMPC group was higher than the rejection rates of the comparison groups for each of the most unfair offers ($7/$3, $8/$2, $9/$1). These results suggest that emotion regulation processes subserved by VMPC are a critical component of normal economic decision making. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2490711/ The Ultimatum Game and brain: a meta-analysis of neuroimaging studies. Gabay et al https://www.ncbi.nlm.nih.gov/pubmed/25454357 Neuroscience and Biobehavioral Reviews 47 (2014) 549–558 Quantitative summary of neuroimaging studies in social decision-making with a metaanalysis of 11 fMRI studies of the UG, including data from 282 participants. Consistent activations in the anterior insula, anterior cingulate cortex (ACC), supplementary motor area (SMA) and cerebellum in response to unfair offers. Robust activation in the ACC, SMA and putamen were seen when deciding to reject rather than accept UG offers. These are consistent with models of motivational conflict during the UG decision-making process, a response to norm violations, with a possible role for the reward system. WHY WOULD WE HAVE PREFERENCE for suffering financial losses rather than accepting unfair divisions ? DICTATOR GAME: Larger assault on homo oeconomicus selfish behavior. You get some money. You can keep it all. Or you can share with someone. Use UG protocol but the other person cannot veto your decision. Meta-analysis (Engel) reports average person gives 28% to other person.
Studies that explore the giving decision:
Schier, Ockenfels, Hofmann,Moral values and increasing stakes in a dictator game Journal of Economic Psychology: Volume 56, October 2016, Pages 107-115 Using representative US sample (N = 1519), we compare hypothetical moral fairness values from the Moral Foundations Sacredness Scale withbehavior in an incentivized dictator game with either low or high stakes. All received endowment of 10 tickets and were asked how many tickets they would share with an anonymous co-player. Tickets could be used as entry tickets to an online raffle. Chance of ticket winning was 0.13%. So $10 worth <1 cent and $500 worth 65 cent. Study finds People with high moral fairness values fail to live up to their high fairness standards when stake size increases. “The Devil – the money-- made me do it.” “Individual differences in good manners rather than compassion predict fair allocations of wealth in the dictator,game:” (Zhao Ferguson Smillie, Journal of Personality) Motivations driving giving anything may represent either emotional concern for others (compassion), adherence to social norms regarding fairness (politeness), or both –“good manners” or adherence to norms concerning fairness,
Achtziger, A., Alós-Ferrer, C., & Wagner, A. K. (2015). Money, depletion, and prosociality in the dictator game. Journal of Neuroscience, Psychology, and Economics, 8(1), 1–14 Nondepleted dictators initially resist the tendency to act selfishly, but eventually become depleted or learn to act selfishly. Hence, pro-social behavior may be short-lived, and ego depletion uncovers the default tendencies for selfishness.
Some studies analyze allocation of time instead of money. “Link between Honesty-Humility and Dictator Game giving Isabel Thielmann , Benjamin E. Hilbig Journal of Research in Personality. Jouxtel, J., Voluntary Contributions of Time: Time-based incentives in a linear public goods game, Journal of Economic Psychology (2019). Volume 75, Part A, December 2019, finds similar patterns of giving and depletion over time.