Costly Signals and Cooperation Károly Takács and András Németh MTA TK Lendület Research Center for Educational and Network Studies (RECENS) and Corvinus University of Budapest New Developments in Signaling and Game Theory Monte Verità, Switzerland, October 14-19, 2012 Support: Hungarian Scientific Research Fund (OTKA), PD 76234.
Costly signaling theory Job market signaling (Spence, 1973) Selection for a handicap (Zahavi, 1975) Unobservable quality can be signaled efficiently if the cost of producing the signal is lower for high quality individuals than for low quality individuals Most interesting case: when interests diverge 2
Hawks and Doves cooperate defect T > R > S > P cooperate R, R S, T defect T, S P, P wc = zr + (1 - z)s wd = zt + (1 - z)p fitness (payoff) Unique ESS: is a mixed strategy equilibrium (or in the domain of pure strategies the unique evolutionary stable state is a mixture of hawks and doves) S P equilibrium Pre-play signaling: Maynard Smith and Parker, 1976; Maynard Smith, 1994; Kim, 1995; Számadó, 2000 w D w C T R z 3
The Prisoner s Dilemma cooperate defect T > R > P > S cooperate R, R S, T defect T, S P, P fitness Defection is dominant strategy Unique ESS: defection wc = zr + (1 - z)s wd = zt + (1 - z)p w D T R What pre-play signals could be used for: A, selective play B, signal cooperative intentions P S w C z 4
Signals do not help cooperation Costly signaling in the single-shot PD makes no sense, because defection is a dominant strategy Even if the other party signals cooperative motives, the best reply is defection Signaling therefore is just costly and does not reveal crucial information Despite its uselessness, pre-play communication increases cooperation in some behavioral experiments 5
Game: PD with pre-play signaling C D C R S D T P SCC SCD SDC SDD ncc ncd ndc ndd cost SCC R R S S R R S S s C SCD R R S S T T P P s C +r T>R>P>S SDC T T P P R R S S s C +r SDD T T P P T T P P s D ncc R S R S R S R S - ncd R S R S T P T P r ndc T P T P R S R S r ndd T P T P T P T P - Simultaneous pre-play decisions: signal (S), no signal (n) Choices are observed and then the PD is played There is a non-zero cost of monitoring signals (r) Costs decrease payoffs and s D >s C >0, r>0 6
Signals do not help cooperation C D C R S D T P SCC SCD SDC SDD ncc ncd ndc ndd cost SCC R R S S R R S S s C SCD R R S S T T P P s C +r T>R>P>S SDC T T P P R R S S s C +r SDD T T P P T T P P s D ncc R S R S R S R S - ncd R S R S T P T P r ndc T P T P R S R S r ndd T P T P T P T P - ndd-ndd is a Nash-equilibrium no single mutant can invade (ndd does strictly better): ndd is ESS 7
End of the story? C D C R S D T P SCC SCD SDC SDD ncc ncd ndc ndd cost SCC R R S S R R S S s C SCD R R S S T T P P s C +r T>R>P>S SDC T T P P R R S S s C +r SDD T T P P T T P P s D ncc R S R S R S R S - ncd R S R S T P T P r ndc T P T P R S R S r ndd T P T P T P T P - ndd is a best response to all strategies except to conditional cooperation there are two mixed strategy Nash-equilibria! 1. SCD ndd 2. SCD SDC ndd or SCD SDD ndd 8
The possibility of cooperation Mixed ESS: SCD ndd proportion π of SCD types in ESS: π SCD = + r R P The costlier the signal is, the better for conditional cooperation! (in this ESS) Mixed ESS: SCD SDC ndd or SCD SDD ndd s C 9
The possibility of limit cycles 2 Limit cycles: SCD SDC ndd SCD or SCD SDD ndd SCD attractors from a wide range of starting populations Example: starting from equal proportions And modifying signaling costs, T=5, R=3, P=1, S=0 Note: figures are not exact, just approximations, red area covers equilibria or limit cycles including conditional cooperation Monitoring costs: 0.01 0.05 0.1 s D ndd s D 0 1 s C s C 10
100% 90% 80% 70% 60% 50% SCC SCD SDC SDD ncc ncd ndc ndd Mutation could favor cooperation 40% 30% 20% 10% 0% 100% 90% 80% 70% SCC SCD SDC SDD ncc ncd ndc ndd T=5, R=3, P=1, S=0, s C =0.1, r=0.1, s D =1, equal starting proportions, mutation rate: 0.005, 0.01, 0.05 SCD 60% 50% 40% 30% 20% 10% 0% SDD ndd 100% 90% 80% 70% SCC SCD SDC SDD ncc ncd ndc ndd 60% 50% 40% 30% 20% 10% 0% 11
Structured interaction Extension: structured populations interactions take place within a spatial proximity d as cooperative intentions cannot be learnt for all possible interaction partners, signaling might also be important in these situations random interactions is a limiting case with d we avoid memories and repeated interactions we assume a complete generation change in a fixed population size N after one round of interactions agents are selected sequentially and matched with a partner within d payoffs then are averaged according to the number of interactions and mean fitness values are calculated strategies with fitness values that allow for less than one offspring become extinct offspring are born within a distance of δ of a randomly selected individual of a given genotype to an empty location 12
The effect of interaction range d Mean final proportions of different genotypes after 400 generations, all starting from equal proportions of the eight types randomly scattered in space. Notes. 3096 runs, T=5, R=3, P=1, S=0 and s D =0.2, s C =0.1, r=0.1, 40% density of the space, the reproduction range is varied with values of {5, 10, 15, 20, 25}. interaction range Mean final proportions of different genotypes after 500 generations, all starting from equal proportions of the eight types randomly scattered in space. Notes. 500 runs, T=5, R=3, P=1, S=0 and s D =1, s C =0.1, r=0.1, 40% density of the space, a single interaction per generation, the reproduction range is 5 interaction range 13
Population density favors defection population density Mean final proportions of different genotypes after 350 generations, all starting from equal proportions of the eight types randomly scattered in space. Notes. 2500 runs, T=5, R=3, P=1, S=0 and s D =0.2, s C =0.1, r=0.1, 40% density of the space, the reproduction range is 5, the interaction range is varied with values of {5, 10, 15, 20, 25}. 14
The effect of signaling costs s D Mean final proportions of different genotypes after 600 generations, all starting from equal proportions of the eight types randomly scattered in space. Notes. 2500 runs, T=5, R=3, P=1, S=0 and s C =0.1, r={0.01, 0.02, 0.03, 0.05, 0.1}, 40% density of the space, the reproduction range is 5, d=5. r Mean final proportions of different genotypes after 600 generations, all starting from equal proportions of the eight types randomly scattered in space. Notes. 2500 runs, T=5, R=3, P=1, S=0 and s C =0.1, 40% density of the space, d=5, the reproduction range is 5, and s D is varied with a values of {0.15, 0.2, 0.3, 0.5, 1}. 15
Summary Pre-play signaling is not hopeless in the PD Honest signalers will never be alone, they will be chased or complemented by deceitful signalers and unconditional defectors Signaling costs largely determine the range of possible equilibria Mutation can favor conditional cooperation In spatial interaction, population density favors defection There is an optimal value of interaction range, which depends on signaling costs it means that interaction range has a non-linear effect 16
Questions for experiments Can costly pre-play signaling increase cooperation in the PD? Do we see more cooperation if signaling costs are higher for defectors, than if costs are the same for everyone? Will fake signaling prevail? Are players interested in reading signals at all in the PD, if it is costly? Do we see more cooperation and signaling in structured interaction, than in random interaction? Do costly signals increase cooperation, if interaction is structured? Do we see more cooperation in structured interaction if signaling costs are higher for defectors, than if costs are the same for everyone? What are the best conditions that can help to maintain high level of cooperation over time in the PD? 17
Method We first categorize subjects as cooperators and defectors based on their play in a simple PD 2x2 design: 1. We introduce signals at a cost that is either differentiated or not for cooperators and defectors 2. PD with random reshuffling or PD with structured interaction 18
PD with pre-play signaling L R L 1000, 1000 0, 1250 R 1250, 0 250, 250 HUF rewards (1000 HUF = approx. 4 ) +900 HUF participation fee, average payment: 1300 HUF signaling cost: 100 HUF signaling cost for defectors, if differentiated: 200 HUF signal reading cost: 10 HUF 19
Subjects 8 sessions, N=160, 40 in each condition structured interaction: random pair from 6 neighbors October 2012, Corvinus University of Budapest 100% answered quiz questions correctly about the baseline PD (everyone understood the basic task perfectly) only 1 subject out of 160 did not answer all quiz questions correctly about signaling 20
Results Baseline cooperation rate: 46.9% Cooperation rate in first round with pre-play signaling: 43.8% change is not significant 21
Cooperation in different conditions cooperation: no difference between the conditions 22
Signaling over time 23
Sending and receiving signals in different conditions 24
Are there cooperator types at all? Cooperators do cooperate more 25
Honest and fake signaling But cooperators do not signal more!!! Also: receiving choice does not differ 26
Does the theory work after all? 27
Summary Costly pre-play signaling does not increase cooperation in the PD We do not see more cooperation if signaling costs are higher for defectors, than if costs are the same for everyone Players were interested in reading signals, this interest naturally declined when less signals were sent Most signaling activity: with equal costs and structured interaction! But this is because of pooling: everyone signals! Signaling works as expected only if: costly signals and structured interaction! Fake signaling is dominant (counter-value?) if random interaction and equal costs 28
Related work: Németh A. and Takács K. 2007. The Evolution of Altruism in Spatially Structured Populations. Journal of Artificial Societies and Social Simulation, 10(3): 4. Németh A. and Takács K. 2010. The Paradox of Cooperation Benefits. Journal of Theoretical Biology, 264: 301-311. e-mail: karoly.takacs@uni-corvinus.hu http://www.uni-corvinus.hu/~tkaroly 29