SN- Lecture 12
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Network games model strategic interactions between players on a network. In these games, players care about the actions of their neighbors and payoffs depend on both a player's own action and those of their neighbors. Three key aspects of network games are: games with strategic complements where players' payoffs increase as more neighbors take an action; games with heterogeneous players who have different preferences over actions; and games with endogenous link formation where players strategically form connections. An experiment on network games found that when player preferences conflict, networks tend to segregate based on preferences and players aim to coordinate within their segregated subgroups rather than achieve the socially optimal outcome.
SN- Lecture 12
- 1. Games On NetworksGames On Networks Lecture 12
- 2. Aim Lecture 12 To understand how networks & games are related Games with strategic complements Games with heterogeneous players Games with endogenous link formation An experiment on network games
- 3. So far... We are interested in understanding networks & behavior Now by bringing strategic interaction into play Network Games
- 4. In network games Decisions to be made Not just a simple diffusion or contagion process People care about what other people are doing Complementarities I want to buy/use a certain software only if other people are buying/using that same software Languages, attending to social events... Interdependencies between individuals
- 5. For this purpose We use game theory as a tool to try and understand how behavior relates to network structure
- 6. Players on a network Different individuals each making decisions Games on Networks Care about the actions of neighbors Interdependencies in payoffs (i.e., their utility function) Our Focus What can we say about behavior & how it relates to network structure
- 8. Each player chooses an action xi in {0,1} Network game Either use the software or not Payoff will depend on my choice the choice of my neighbors ui(xi, xNi) How many choose 1 or 0Whether I choose 1 or 0
- 9. Either take an action or not simplifying assumptions 1 2 Care about actions of neighbors but not who they are (Identities) 3 Fixed networks 1 or 0, but not a wide range of choices I don’t have best friends or closer neighbors I cannot choose my neighbors but they are given It will become richer soon...
- 10. Complements vs. Substitutes Types of games As more of my friends take an action, it is more attractive to me ui(1,m)-ui(0,m)≥ ui(1,m’)-ui(0,m’) for all m≥m’ Strategic complements - Increasing differences Strategic substitutes - Decreasing differences As more of my friends take an action, it is less attractive to me ui(1,m)-ui(0,m)≤ ui(1,m’)-ui(0,m’) for all m≥m’ Coordination Anti-Coordination
- 11. Example 1 An agent chooses action 1 if at least two neighbors do: threshold of one. Payoff action 0: ui=0 Payoff action 1: ui=-1, if less than two neighbors choose 1 ui=1 per coordination, if at least two neighbors choose 1 Remember: payoffs are ordinal. The number doesn’t matter
- 12. Example 1 For example: I only choose to use a technology (choose 1) if at least 2 of my friends are using it, otherwise I rather not do it. An agent chooses action 1 if at least two neighbors do: threshold of one. 0 0 0 0 00 0 0 0 0 0 Only one neighbor, so they can’t choose 1
- 13. Example 1 For example: 0 0 0 0 00 0 0 0 0 0But, how about this guy? I only choose to use a technology (choose 1) if at least 2 of my friends are using it, otherwise I rather not do it. An agent chooses action 1 if at least two neighbors do: threshold of one.
- 14. Example 1 For example: I only choose to learn to play bridge if at least 2 of my friends know how to play it. otherwise I rather not do it. 0 0 0 0 01 0 0 1 1 0 Each has at least two friends choosing 1 An agent chooses action 1 if at least two neighbors do: threshold of one.
- 15. Nash Equilibrium No player wants to change her behavior alone, fixing what her neighbors are doing 0 0 0 0 00 0 0 0 0 0 Case 1: The technology is never used
- 16. Nash Equilibrium Case 2: These three people adopted the technology because each of them has 2 neighbors using it 0 0 0 0 01 0 0 1 1 0 In this case no one else would want to do it
- 17. Network Games Heterogeneous Preferences With & Strategic Complementarities
- 18. Either take an action or not simplifying assumptions 1 2 Care about actions of neighbors but not who they are (Identities) 3 Fixed networks 1 or 0, but not a wide range of choices I don’t have best friends or closer neighbors I cannot choose my neighbors but they are given
- 19. Strategic complementarities Games with complements The choice to take an action by my friends increases my relative payoffs to taking that action (tipping point) Tipping point: Threshold (ti) Number of my neighbors adopting the technology (choosing action 1) 0 niti ------- I choose 0 ------- ------- I choose 1 ------- if ni(1)<ti I choose 0 & if ni(1)≥ti I choose 1
- 20. Education decisions: university, human capital? Some examples of SC Increases access to jobs (the more people you know who are well educated the higher the chances) Invest if at least k neighbors do the same Technology adoption/ learn a language Peer influence were the pressure grows with the size of the crowd Cheating, doping Anti-social behavior: smoking among teens If you are in sports and others are taking “performance enhancing drugs” The relative payoff goes up the more others are doing it... even though the more people do it doesn’t make it better for an athetle
- 21. Example 2 Players have preferences for the different options: like one more than the other any two players are identicalSo far: Heterogeneous Network Games: Conflicting Preferences with P. Hernández & A. Sánchez (GEB, 2013) Types: ui= a for 1 & b for 0 ui= b for 1 & a for 0 a>b
- 22. Example 2 1 > 0 0>1 2a,2a a,b b,a 2b,2b 1 0 1 0 2a,2b a,a b,b 2b,2a 1 0 1 0 2b,2b b,a a,b 2a,2a 1 0 1 0 Conflicting preferences: Players want to be together, but have different opinions about which is the most desirable outcome
- 23. Example 2 An agent chooses her favorite action if ti=1/3 of her neighbors choose it Neighbors choosing 1 0 ni ti ti 0 0 1 0 1 1 There are two thresholds: One to choose what I like (ti) & one to choose what I dislike (ti)
- 24. With heterogeneity We can understand & model the strength of social influence needed to make players adopt certain behaviors Education decisions: The effect is different for people who would prefer to study than for those who would prefer to jump into the job market Anti-social behavior: Kids who would like smoking need less influence to start than others
- 25. Example 2 We know need to consider: preference (type) & behavior (action chosen) 00 00 00 00 0000 00 00 00 00 00 The first digit is the preference & the second the behavior
- 26. Example 2 We know need to consider: preference (type) & behavior (action chosen) Case 1: This is a Specialized satisfactory equilibrium. 00 00 00 00 0000 00 00 00 00 00 All choose the same action & are happy
- 27. Example 2 We know need to consider: preference (type) & behavior (action chosen) 11 11 11 11 1111 11 11 11 11 11 All choose the same action & are happy Case 1: This is a Specialized satisfactory equilibrium.
- 28. Example 2 We know need to consider: preference (type) & behavior (action chosen) Case 2: This is a Specialized frustrated equilibrium. 00 00 00 00 0010 10 10 10 10 00 All choose the same action but the 1’s are frustrated
- 29. Example 2 We know need to consider: preference (type) & behavior (action chosen) Case 2: This is a Specialized frustrated equilibrium. 01 01 01 01 0111 11 11 11 11 01 All choose the same action but the 0’s are frustrated
- 30. Example 2 We know need to consider: preference (type) & behavior (action chosen) Case 3: This is a Hybrid satisfactory equilibrium. 00 00 00 00 0011 11 11 11 11 11 All choose the preferred action (happy) & both actions coexist
- 31. Example 2 We know need to consider: preference (type) & behavior (action chosen) Case 4: This is a Hybrid frustrated equilibrium. 00 00 10 00 0001 11 11 11 11 01 Some choose the non-preferred action & both actions coexist
- 32. What does this mean? People’s relationships with others determine the benefits they can get and the goals they can achieve 1 Marsden & Gorman, 2001 About 2/3 of the working population in western industrialized societies (informal social ties) Participation in political protest Affected by friendship and family networks Opp & Gern, 1993 Finding jobs
- 33. But... 2 People’s characteristics determine the social relationships they form Opportunity (Contact theory) More of a chance of meeting your own type The possibility that you meet people could be biased by attributes (i.e, race) Costs & benefits Social pressure or social competition Common attributes (i.e., language, culture, knowledge) make it easier
- 34. Network Games Endogenous link formation With & Strategic Complementarities Heterogeneous Preferences &
- 35. Either take an action or not simplifying assumptions 1 2 Care about actions of neighbors but not who they are (Identities) 3 Fixed networks 1 or 0, but not a wide range of choices I don’t have best friends or closer neighbors I cannot choose my neighbors but they are given
- 36. A person’s social network promotes her goal achievement & 1 2 A person will invest in her social network (i.e., form relationships) depending on its instrument value Combined arguments What if people’s preferences on the possible outcomes are in conflict? Heterogeneity in preferencesHeterogeneity in preferences
- 37. The Model 2 stage network game Nodes: Purposive rational actors Players Type ✓i 2 {0, 1} N = {1, . . . , n} Affiliation Each player announces who they wish to link with pi = (p1 , . . . , pn ) A link forms iff both players propose to each other Behavior Adoption Players observe the network and choose an action X = {0, 1} A player of type 1 prefers action 1 over 0
- 38. The Model The network generates payoffs to the players ui(✓i, (p1 , . . . , pn ), xi, xki(g)) = ✓i xi (1 + Pki j=1 I{xj =xi}) cpi Indicator function coordination Type parameter a if happy b if frustrated Linking cost for every proposal a>b>c Coordination game with strategic complementarities
- 39. The Model Equilibrium examples Subgame perfect Nash equilibrium No links with uncoordinated neighbors are kept Only links with coordinating neighbors are part of an equilibrium No unreciprocated links are proposed There are multiple networks that satisfy these properties The more neighbors coordinating with the better Threshold model (tipping point to choose favorite action)
- 40. The effect of different levels of conflict in the preferences (microlevel) on the emerging network configurations (macrolevel) Experimental Design 3 treatments No conflict (15-0) - 30 subjects+ + Low conflict (12-3) - 45 subjects + High conflict (8-7) - 45 subjects One-shot + 15 players + 20 rounds + z-tree Undergraduate students Universitat de Valencia (Spain)
- 41. Experimental Design Multiple equilibria Link Proposal Action 1 socially optimal Complete network Favorite behavior of the majority
- 42. Results Who is proposing connections to whom? Subjects segregate between types (preferences) in the relationships they form 1 Maj proposes & connects most often in No and least often in High+ Min proposes & connects most often in High and least often in Low In High, Min sends more proposals between groups than Maj + + 2 components in heterogeneity
- 43. Results Who is proposing connections to whom? Box plots of proposals & connections within & between groups
- 44. Results Subjects attempt to maximize payoffs within the equilibrium they choose2 How is the level of connectivity? Differences between treatments and groups are systematic+ ANOVA: (Ftreatment=5.00, p<0.01; Fgroup=79.92, p<0.001) + Min networks are denser than the Maj networks + Density is lower in Low than in No + Max. density is reached for Min in High
- 45. Results Box plots of density by treatment and groups How is the level of connectivity?
- 46. Results Subjects reach stable configuration in homogeneity or minorities in heterogeneity3 Is there any stable subgroup & when? NO: complete density from round 12 on LOW & HIGH: Do not reach complete density + lower density than Min Minority LOW: complete density from round 4 on. HIGH: start off with higher densities + complete density from round 6 Once complete density is reached it remains very stable. Majority
- 47. Results Is there any stable subgroup & when?
- 48. Is there any stable subgroup & when? Results
- 49. Is there any stable subgroup & when? Results
- 50. Implications Conflicting preferences social optimality is not reached (risk- dominant equilibrium) Subjects aim for the payoff dominant equilibrium within their segregated Eq. Individual preferences are more focal than payoffs Game theoretic Lit: exogenous networks (risk-dominant equilibrium) endogenous networks (payoff dominant equilibrium) Existence of conflict (not level) leads to segregation Social identity theory, homophily, etc. Further: Less individualistic societies?
- 51. Checklist We have learned that network games can represent multiple social & economic problems Actors relations in the network affect their behavior Actors preferences influence the relationships they form Experimentally, individual characteristics are more salient than payoffs
- 52. Questions?