Costly Signals and Cooperation - ETH Zürich · PDF fileCostly Signals and Cooperation ......

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.

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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

Takács, Németh – Costly Signals and Cooperation

3

Hawks and Doves T > R > S > P

wC = zR + (1 - z)S wD = zT + (1 - z)P

cooperate defect

cooperate R, R S, T

defect T, S P, P

T

R

S

P

z

equilibrium

fitness (payoff)

wD

wC

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)

Pre-play signaling: Maynard Smith and Parker, 1976; Maynard Smith, 1994; Kim, 1995; Számadó, 2000


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The Prisoner’s Dilemma T > R > P > S

wC = zR + (1 - z)S wD = zT + (1 - z)P

cooperate defect

cooperate R, R S, T

defect T, S P, P

Defection is dominant strategy Unique ESS: defection

What pre-play signals could be used for:

A, selective play B, signal cooperative intentions

T

R

P

S

z

fitness

wD

wC


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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


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Game: PD with pre-play signaling C D SCC SCD SDC SDD nCC nCD nDC nDD cost

C R S SCC R R S S R R S S sC D T P SCD R R S S T T P P sC+r

T>R>P>S SDC T T P P R R S S sC+r

SDD T T P P T T P P sD

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 sD>sC>0, r>0


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Signals do not help cooperation

C D SCC SCD SDC SDD nCC nCD nDC nDD cost C R S SCC R R S S R R S S sC D T P SCD R R S S T T P P sC+r







• nDD-nDD is a Nash-equilibrium • no single mutant can invade (nDD does strictly better):

nDD is ESS


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End of the story?

C D SCC SCD SDC SDD nCC nCD nDC nDD cost C R S SCC R R S S R R S S sC D T P SCD R R S S T T P P sC+r







• 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


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The possibility of cooperation

• Mixed ESS: SCD – nDD proportion π of SCD types in ESS:

The costlier the signal is, the better for conditional cooperation! (in this ESS) • Mixed ESS: SCD – SDC – nDD or SCD – SDD – nDD

PRrsC

SCD −+

=π


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The possibility of limit cycles • 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

sD nDD sD

sC sC 0 1

2

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Mutation could favor cooperation

T=5, R=3, P=1, S=0, sC=0.1, r=0.1, sD=1, equal starting proportions, mutation rate: 0.005, 0.01, 0.05

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%SCCSCDSDCSDDnCC

nCDnDCnDD

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%SCCSCDSDCSDDnCC

nCDnDCnDD

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%SCCSCDSDCSDDnCC

nCDnDCnDD

nDD

SCD

SDD


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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

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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 sD=0.2, sC=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 sD=1, sC=0.1, r=0.1, 40% density of the space, a single interaction per generation, the reproduction range is 5

interaction range Takács, Németh – Costly Signals and Cooperation

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Population density favors defection

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 sD=0.2, sC=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}.

population density


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The effect of signaling costs 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 sC=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.

sD

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 sC=0.1, 40% density of the space, d=5, the reproduction range is 5, and sD is varied with a values of {0.15, 0.2, 0.3, 0.5, 1}.

r


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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


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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?

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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


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PD with pre-play signaling HUF rewards

(1000 HUF = approx. 4 €)

+900 HUF participation fee,

average payment: 1300 HUF

L R

L 1000, 1000 0, 1250

R 1250, 0 250, 250

signaling cost: 100 HUF signaling cost for defectors, if differentiated: 200 HUF

signal reading cost: 10 HUF


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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


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Results

• Baseline cooperation rate: 46.9% • Cooperation rate in first round with pre-play signaling: 43.8% • change is not significant


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Cooperation in different conditions


cooperation: no difference between the conditions

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Signaling over time

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Sending and receiving signals in different conditions


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Are there cooperator types at all?


Cooperators do cooperate more

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Honest and fake signaling


But cooperators do not signal more!!! Also: receiving choice does not differ

27 Takács, Németh – Costly Signals and Cooperation

Does the theory work after all?

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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


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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: [email protected] http://www.uni-corvinus.hu/~tkaroly


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