Game Theory and SocietyModels of Social Interaction in Sociological Research

ETH Zurich, July 27-30, 2011

Game Theory and Society

“The aim of the conference is to explore the potential of game theory for sociological theory and its application to sociological research broadly considered.

For this purpose the conference brings together scholars with different disciplinary backgrounds to focus on topics of game theory relevant to sociology and society at large.”

Modern Game Theory

► Repeated games► Incomplete/asymmetric information► Signaling games► Evolutionary game theory (learning, imitation,

bounded rationality, non-linear dynamics)► Behavioral game theory (social preferences,

fairness, altruism, envy etc.)► Games and social networks, spatial games► Agent based simulation► Experimental approach

► In the last two decades key questions of sociology on:

• Social order• Cooperation • Trust• Reciprocity• Social capital• Social norms and sanctions etc.

were explored by economists, other behavioral scientists but only a minority of sociologists.

► Ironically, in the recent past there was more „experimental sociology“ in economics than in sociology.

► However, the focus of research is human behavior in situations of strategic interaction. This is a highly interdisciplinary research program.

• Herbert Gintis (2000): „ ...game theory is a universal language for the unification of the behavioral sciences. Games are about life and are increasingly recognized as basic to biology, anthropology, sociology, political science, psychology, economics, and the other behavioral sciences.”

Semper Aula ETH-Zurich, August 2008

► Clearly, the rationality assumptions of game theory are often violated.► Nevertheless, classical game models serve as a point of reference. ► And the interesting question is how to explain the discrepancy between rationality predictions and actual behavior.► Evolutionary processes, bounded rationality, learning, imitation, and social preferences are the elements of more realistic theories of human behavior.► These models have to be examined in experiments, field experiments or „natural experiments“.Here is an example:

• Golden balls

http://www.youtube.com/user/GoldenBoIIocks

This video contains many elements of behavioral game theory:

• High stakes• Communication,• cheap talk (partially) matters• Promise kept (and broken)• Inequality aversion• Perceived intentions• Heterogeneous preferences• 50 % cooperation

„Golden balls has taught me that Game Theory has applications in real life.“

olgepwnt, one of 3980 comments to the youtube video clip

Emergence of Social Norms AmongHeterogeneous Actors

Andreas Diekmann and Wojtek PrzepiorkaETH Zurich

Game Theory and SocietyETH Zurich, July 27-30, 2011

• Many studies are concerned with the stability of existing norms. Here, we want to study the emergence of new social norms.

• With heterogeneous actors (compared to symmetric situations of strategic interaction).

• Basic game is a volunteer‘s dilemma: In this game there is both a problem of coordination and conflict of interest.

• Key question: Does heterogeneity promote cooperation?

• 3 experiments with the repeated symmetric and asymmetric dilemma. (Out of a series of 5 experiments conducted 2008 to 2011)

Volunteer’s DilemmaThe Volunteer’s Dilemma is a N-player binary choice game (for N ≥ 2) with a step-level production function. A player can produce a collective good at cost K > 0. When the good is produced, each player obtains the benefit U > K > 0. If no player volunteers, the good is not produced and all players receive a payoff 0.Example: “Bystander intervention in emergencies” (Darley & Latane 1968)

N-Player Game:

Formal Models & Extensions,e.g.: Asymmetric VOD (Diekmann 1993, Weesie 1993) Timing and incomplete information in the VOD (Weesie 1993, 1994) Cost sharing in the VOD (Weesie and Franzen 1998) Quantal response and social preferences (Goeree and Holt 2005) Evolutionary Analysis of the VOD (Myatt and Wallace 2008) Dynamic Volunteer’s Dilemma (Otsubo and Rapoport 2008) Bounded Rationality VOD (Tutic 2011)

0 1 2 … N - 1 Other C-Players

C U - K U - K U - K … U - K

D 0 U U … U

No dominant strategy, but N asymmetric pure equilibria in which one player volunteers while all other defect. Prediction: Defection ► q = N-1√ K/U“Diffusion of Responsibility” derived from a game model

„Diffusion of Responsibility“

• „Bystander intervention in emergencies“ (Darley and Latane 1968)•Investment in R & D or imitating competing firm`s innovations (Eger et al. 1993)•The quality of a seminar is sometimes low if two or more professors shareresponsibilities.

• Sanctioning dilemma. If a single person is sufficient to punish a norm violation.•E-Mail help requests (Barron and Yechiam 2002)•Two or more „privileged actors“. Olson (1965) ,“Logic of collective action“.•Committee with veto. Expecting a colleague‘s veto if a decision is unpopular.•Alarm calls of animals if a predator is approaching (Archetti 2009, 2010)• Computer networks: „Backbone construction in selfisch wireless networks“(Lee et al. 2007)

•„ Mahmihlapinatapai“, a single word for a volunteer‘s dilemma in Tierrade Fuego (Rapoport 1988)

Examples of VOD

Experiment I with repeated symmetric volunteer‘s dilemma, U = 80, K = 50(0.80 versus 0.30 CHF per round, partner matching),

Group size N = 2, 3, 5

Emergence of Social Norms in the Repeated Symmetric Dilemma GameEffect of Group Size

► Complexity of Cooperation: Only small groups succeed to establish „take-turn“norms.

► Social learning is important (long sequences of repeated interactions).

Group size N = 2 N = 3 N = 4% take turn norms

56.5 22.9 4.7

Asymmetric Volunteer’s Dilemma• Heterogeneity of costs and gains: Ui, Ki for i = 1, …, N; Ui > (Ui – Ki) > 0

• Special case: A “strong” cooperative player receives Us – Ks , N-1 symmetric “weak” players’ get U – K whereby:Us – Ks > Uw – Kw

• Strategy profile of an “asymmetric”, efficient (Pareto optimal) Nash equilibrium:

s = (Cs, D, D, D, D, …,D)

• i.e. the “strong player” is the volunteer (the player with the lowest cost and/or the highest gain). All other players defect.► In the asymmetric dilemma: Exploitation of the strong player by the

weak actors.►Paradox of mixed Nash equilibrium: The strongest player has the

smallest likelihood to take action!

Cooperation:

The Greek polis’ had been under threat by the Persian emperor Darius in the fifth century B.C.

The strategic interactions of about 60 polis resemble an asymmetric volunteer’s dilemma. Most of them defected while Athens, the most powerful state, was almost alone to act in the common interest.

The collective good of Greek independence was preserved by the victory of Athens at Marathon 490 B.C.

Note: In modern times strong actors of military alliances do not necessarily produce common goods. Often, the outcome is a public bad!

Here is a historical example from ancient Greece(see Whelan 1997):

Emergence of Social Norms in an Asymmetric Dilemma(Experiment 2)

Group size N = 3, series of interactions with 48 to 56 repetions,120 subjects

1. Symmetric VOD (control): U = 80, K = 50

2. Asymmetric 1: Weak players: U = 80, K = 50,Strong player: U = 80, K = 30

3. Asymmetric 2: Weak players: U = 80, K = 50,Strong player U = 80, K = 10

For all players the collective good has the same value U = 80.Strong players have lower cost to produce the collective good.

Dynamics of Behavior in the Asymmetric Game

Strong player cooperates

Strong player cooperates

Trial and error withtake turn norm

Overall rate of cooperation Efficient cooperation

► Take turn norms in the symmetric game► Asymmetric game: Strong actors cooperate more frequently than weak actors.► However, no efficiency gain in the asymmetric situation for group size N = 3,

but likely for larger group size► Experiment 4: „Stranger matching“ instead of „partner matching“. Clear advantage

of asymmetric interaction. Also, the asymmetric game is much more efficient.

Sanctioning as a Volunteer‘s Dilemma

Instead of a public good game with punishment:• A single observer is sufficient to sanction a norm

violation.• Sanctioning is costly (K) and restores the social

norm. Both, sanctioning and free riding victims of a norm violation profit (U).

• Groups of four. A sender allocates a certain amount of money to three receivers. He or she can choose a fair or unfair allocation rule. If the allocation is unfair receivers have the right to veto (sanction). In more detail:

Sanctioning as a Volunteer‘s Dilemma(Experiment 3)

Symmetric Asymmetric Symmetric,binary choice

SenderTransfer

0 – 400 Rp.(VOD: Senderkeeps > 310)

0 – 400 Rp.(VOD: Senderkeeps > 310)

0 or 240 Rp. (VOD: Sender keeps 400)

Recipientsanctioncost

K1,2,3 = 50K1,3 = 50;K2 = 40 K1,2,3 = 50

108 subjects, groups of N = 4: 1 sender, 3 recipients, 24 rounds, stranger matching, in every round role of sender or recipient assigned by random. Recipients had an endowment of 80 Rp., sender decides whether to share 400 Rp. A fair sender keeps 160 and sends 240 (fairness norm).Round 1 - 8: Recipients had no veto powerRound 9 - 24: Any recipient had the opportunity to reduce sender‘s income to 160 on cost K if the sender had violated the fairness norm.

Example: Sender keeps 340

Veto: Sender‘s payoff is reduced to 160

Receiver‘s payoff is increased by 180 and everyReceiver gains 60

Collective good U = 60

Volunteer‘s cost K = 50

U – K > 0 ► volunteer‘s dilemma

Asymmetric K1,,3 = 50, K2 = 40,Symmetric K1,2,3 = 50

Sanctioning as a volunteer‘s dilemma

Sender keeps more than 310. Veto reduces sender’s incometo 160 and increases every receiver’s income by more than 50.(U > 50, K = 50) U – K > 0 → Volunteer’s dilemma

Cooperation rate if transfer established “Missing HeroDilemma” (volunteer: U – K < 0, other players: U > 0)

Sender keeps less than 310 (U < 50)(asymmetric less than 280, U < 30)U – K < 0 → Altruistic punishment

• Punishment threat increases the level of cooperation in the continous transfer game.

• Limited amount of altruistic punishment.• Series of experiments show: In an asymmetric

game there is a higher level of cooperation by a „strong“ player compared to free riding „weak“ players. The social norm emerges: „A strong player is expected to volunteer“

• In symmetric games „take turn“ norms evolve –but only if group size is small.

• Under specified conditions, heterogeneity enhances the level of cooperation and increases efficiency on cost of the strong players.

• Strong players are exploited by weak players.► Heterogeneity/social inequality concerning resources and interests deserves much more attention in experimental game theory.

The Issue of Deception

• If strong actors loose they have an incentive to deceive.

• Understate your strength!• The (ancient) Greece had sophisticated

institutions to prevent deception.• Moore (1992) reports on the rules how to find

wealthy people paying for the „choregia“, a large festivity and theatre.

Choosing a volunteer to sponsor the choregia in ancient Greece (Moore 1992).

The strongest (wealthiest) citizen is expected to volunteer.

Fellow citizens nominate a person who is assumed to be the richest - say Kroisos.

Kroisos has the right to refuse. Then he has to name a citizen (say Kyros) who is supposed to be wealthier.

However, Kyros may veto. In this case Kyros and Kroisos have to exchange their total wealth and Kroisos will take responsibility for the festivity.

Solution of a „Volunteer‘s Dilemma“ by Emperor Penguins (Dawkins, Selfish Gene)

Wikipedia Commons

„The strong are exploited by the weak“ – but often it is the other way around.

Classical economics:

No problem: The outcome is a Nash equilibrium and it is Pareto-optimal!

Behavioral game theory

•Large stakes•Cheap talk matters•Binding force of a promise•Social preferences•Heterogeneous players•50 % cooperation!

Sender has binary choice, symmetric K1,2,3 = 50

Asymmetric K1,,3 = 50, K2 = 40,Symmetric K1,2,3 = 50

Average sender transfer in first and second part across

treatments

Recipients cooperation rate across treatments and

recipient types (rounds 9 – 24)

Frequency distribution of average sender transfersymmetric

asymmetric symmetric 2 (binary choicegame)

Volunteer’s DilemmaThe Volunteer’s Dilemma is a N-player binary choice game (for N ≥ 2) with a step-level production function. A player can produce a collective good at cost K > 0. When the good is produced, each player obtains the benefit U > K > 0. If no player volunteers, the good is not produced and all players receive a payoff 0.

Prediction: Defection ► q = N-1√ K/U“Diffusion of Responsibility” derived from a game model

Asymmetric Volunteer‘s Dilemma

A „strong“ actor can produce the collectivegood on lower costs than „weak“ players.Us – Ks > Uw – Kw > 0.

Prediction: a) More efficient than symmetric game.b) Exploitation of strong actors.

87 subjects, groups of N=3, 56 rounds, „stranger matching“, U=80, Kw=50, Ks1=30, Ks2=10

Diekmann and Przepiorka 2011

Choose the probability of defection q such that an actor is indifferent concerning the outcome of strategy C and DU – K = U (1 – qN-1)U – K = U - U qN-1

qN-1 = K/UDefection ► q = N-1√ K/UCooperation ► p = 1 - N-1√ K/U

“Diffusion of Responsibility”derived from a game model

Mixed Strategy Nash Equilibrium

p =1- N-1√ K/UIndividual level prediction:„Diffusion of responsibility“.

Here, parsimonious assumptions ofthe simple model are sufficient to derivethis effect and further hypothesis.

Macro level prediction: The probability of collective good production (that at leasta single actor will volunteer) is expected to decrease with group size.However, experiments show that this probability is increasing. The reason is that theModel predictions slightly underestimate the individual tendency to cooperate

Refinements of the model can account for the discrepancy:Goeree and Holt (2005): Quantal response equilibrium and social preferencesTutic (2011) suggests a modification based on assumptions of bounded rationality.

Prob. of cooperation

• A child is in danger of drowning. There are N observers.

► Mixed Nash equilibrium: The guard or the best swimmer has the lowest probability to save the child.

► „Asymmetric“ pure strategy Nash-equilibrium: The best trained swimmer will take action.

Experiment 3: Sequence of one-shot interactions in the symmetric and asymmetric dilemma („stranger matching“, in every interaction the role of the strong player is assigned by random)

87 subjects,groups of N=3,56 rounds

symmetricU = 50, K = 50

asymmetric 1„strong“ K = 30

asymmetric 2„strong“ K = 10