One Step at a Time: Does Gradualism Build Coordination?

1. One Step at a Time: Does Gradualism Build Coordination? Maoliang Ye (Renmin University of China) Sam Asher Lorenzo Casabur Plamen Nikolov April 18, 2013 Shanghai Jiaotong University 1

2. Outline Introduction Experimental Design Results A belief-based learning model with level-k thinking Conclusion 2

3. Introduction • (Tacit) Coordination: attempt to match the actions of others (without knowing what these others will do and without an agreement about what to do) • Coordination in economics and organization (Schelling, 1960; Arrow, 1974) matters for social welfare • Economic development (Ray, 1998; Bardhan, 2005) • International monetary policy (Krugman & Obstfeld, 2009) • Establishment of democracy and rule of law (Weingast, 1997)

4. Introduction • Coordination with Pareto-ranked equilibria • Weakest-link (minimum-effort) coordination • payoff depends on one’s effort and the minimum effort of all group members. • complementary production function (Leontif) • Examples of weakest-link coordination • Flight security, high-speed railway system • Fight against terrorism attack • Dam security • “木桶效应” (“Cannikin Law”)

5. Introduction • Coordination failure is common in the lab (Van Huyck et al., 1990; Knez & Camerer, 1994, 2000; Cachon & Camerer 1996; Weber, 2006; Chaudhuri et al., 2009) • Each individual has seven choices for effort level • For a large group, Pareto-inefficient outcome is attained • Strategic uncertainty even without interest conflict

6. Introduction • How to induce all agents to contribute a high level of effort in coordination? • Mechanisms to facilitate coordination • Complete information feedback (Brandts & Cooper, 2006a) • Communication (Cooper et al., 1992; Charness, 2000; Weber et al., 2001; Duffy & Feltovich, 2002, 2006; Chaudhuri et al., 2009) • Between-group competition (Bornstein et al., 2002; Riechmann & Weimann, 2008) • Voluntary group formation (e.g., Yang et al., 2011) • Gradual increase in group size (Weber, 2006)

7. Introduction • Our potential mechanism: gradually increasing the stake of coordination project • Fixed group size and composition • The short horizon when the group size and group composition have not yet changed • Stake: size of project; cost of participation and potential benefit (if successful) • Gradual increase in the stake of projects

8. Introduction • Examples of gradualism in coordination: • Intra-organization team building • Inter-organization coordination on joint projects • WTO • International arms reduction agreements • Better (cleaner) examples???

9. Introduction • Use a lab experiment to examine the role of exogenous gradualism in coordination within a given group • Behavioral Mechanism Design • Preview of results: gradualism facilitates weakest-link coordination in high-stake projects • Theoretical explanations: A belief-based learning model with level-k thinking

10. Stake Patterns 10

11. Relation with Other Dynamic Coordination/PG Games • Weber (2006) • Romero (2011) • PG game: Offerman and van der Veen (2010) • Dynamic continuous-time public goods provision: Dorsey (1992), Marx and Matthews (2000), Kurzban et al. (2001), Duffy et al. (2007) and Charness et al. (2011). • This study: whether a group should work on smaller tasks before accomplishing a large collective task, rather than whether a call for large collective contributions should be divided into multiple periods to allow accumulations over time.

12. Experimental Design • 2 stages: 12+8 (unknown for subjects) • Partners within stage; strangers across stage • Endowment per period: 20 points • Payoff of each period (weakest-link): • Know current but NOT future stake • Feedback after each period: whether all 4 members contributed, but not how many members contributed (if <4)

13. Hypothesis 1 Multiple equilibria: starting stake level and stake path may affect initial formation and updating on belief about others’ actions Rational players have prior beliefs about others’ actions before the game starts, and update their beliefs according to the outcome in each period A low stake level at the beginning makes it cheaper to attempt coordination in the face of uncertainty gives rational players stronger beliefs that others will contribute so they are more likely to contribute given the risk-averse preference and the weakest-link payoff structure

14. Hypothesis 1 • After success at a stake level, they reinforce their beliefs about the likelihood that others will contribute at the same stake, otherwise they reduce their beliefs • Previous successful coordination at low stakes may not largely affect players’ posteriors on actions at substantially higher stake levels • Hypothesis 1: A low stake at the beginning is more likely to establish the initial successful coordination than a high stake, and a slow increase in stake can (weakly) better maintain the coordination success than a quick increase.

15. Hypothesis 1 • Alternative way of hypothesis formation: cross-game learning (Cooper & Kagel, 2008, 2009) • learn in one game and generalize it to related games • Gradual increase in stake: high similarity in two consecutive periods

16. Hypothesis 2 • Players may learn about group members via coordination outcomes in stage 1 and form their beliefs about the general population. • When they enter stage 2, their beliefs about new group members may be affected by the coordination performances in stage 1. • Hypothesis 2: Those treated in the “Gradualism” treatment is more likely to contribute when they enter stage 2. And this is driven by its higher success rate at the end of stage 1.

17. Stake Pattern in Stage 1

18. Main Result in Stage 1 • The Gradualism treatment significantly outperforms alternative treatments in the high-stake periods: starting at a low stake and growing slowly lead to • Higher success rate of coordination • Higher earnings 18

19. Group Success Rate Wilcoxon-Mann-Whitney tests on average success rate over periods 7-12: GR vs. BB: p<0.01 GR vs. SG: p=0.06 GR vs. HS: p=0.09 Observations are at the group level (18 obs. each for GR, BB and SG, 10 for HS) 19

20. Average Individual Earning Wilcoxon-Mann-Whitney tests on individual accumulative earnings over periods 7-12: GR vs. BB: p<0.0001 GR vs. SG: p<0.0001 GR vs. HS: p=0.02 Observations are at the individual level (72 obs. each for GR, BB and SG, 40 for HS) 20

21. Income Effect?

22. Income Effect?

23. Coordination Dynamics in Stage 1 • Pattern 1:The contribution and success rates in period 1 are higher for groups with a lower stake (Wilcoxon-Mann-Whitney test: p<0.0001) • NO income effect: no difference in the contribution rate between “Big Bang” and “High Show-up Fee” (ideally S=C4)

24. Coordination Dynamics in Stage 1 • Pattern 2:Conditional on failed coordination at t, most groups fail with the same or a higher stake at t+1. • Pattern 3: Conditional on successful coordination at t, most groups succeed with the same or a slightly higher stake at t+1, but fewer groups keep successful with a much higher stake at t+1. (GR vs. SG in Period 7: p<0.05)

25. Coordination Dynamics in Stage 1 • Pattern 4: The decrease of contribution rates over periods is mostly driven by members in failed groups; but for the “Semi-gradualism” treatment from period 6 to 7, the decrease is largely driven by members in previously successful groups.

26. Coordination Results of Each Group “Big Bang” Groups

27. Coordination Results of Each Group “Semi-gradualism” Groups

28. Coordination Results of Each Group “Gradualism” Groups

29. Coordination Results of Each Group “High Show-up Fee” Groups

30. Main Result in Stage 2 • Those treated in the Gradualism treatment in stage 1 are more likely to contribute upon entering a new group in stage 2. But this difference quickly disappears. • externality of coordination building (or collapse) across different social groups.

31. A Belief-based Learning Model in Weakest-link Coordination • Belief-based learning • Myopic • Adopted by many learning models • Focus on the belief-based learning • Level-k thinking (e.g., Costa-Gomes et al., 2001; Costa-Gomes & Crawford, 2006; Ho et al., 2012; Crawford et al., 2013) • Level-0: nonstrategic • Level-1: best respond to level-0 • Level-k: best respond to level-k-1 • Self-interested, risk averse 31

32. A Belief-based Learning Model in Weakest-link Coordination • N periods of binary weakest-link coordination game with I players • Weakest-link payoff structure • E: endowment • St: stake • α-1: return rate (α>1) 32

33. A Belief-based Learning Model in Weakest-link Coordination • Level-0 player: constant “stake threshold” • A natural extension from the case of continuous playing • Qualitative evidence • Can be rationalized by risk aversion and belief about others • Could be level-1 players in alternative level-k model • Level-k (k>=1) player: belief about level-k-1’s action, best respond, and update the belief 33

34. A Belief-based Learning Model in Weakest-link Coordination • Level-k (k>=1) player i will contribute iff she believes the probability that all her opponents will contribute exceeds a certain value : • Lemma 1: , and • the higher the stake, the higher the “reserved probability of success” which makes a risk-averse rational player willing to have a try 34

35. A Belief-based Learning Model in Weakest-link Coordination Proposition 1: The lower the S1, the (weakly) higher the probability that the coordination at t=1 will succeed. Belief updating for level-1 players: the bound of the minimum of the stake thresholds for all opponents 35

36. A Belief-based Learning Model in Weakest-link Coordination Proposition 2 (Consistency of Success) Conditionalon successful coordination at t with a stake St, the coordination at t+1 will succeed with the same stake (St+1=St). Proposition 3 (Consistency of Failure): Conditionalon failed coordination at t with a stake St, the coordination at t+1 will fail with the same or higher stake (St+1>=St). To explain exceptions in the results: introduce noise in level-0’s playing 36

37. A Belief-based Learning Model in Weakest-link Coordination Proposition 4: Conditional on successful coordination at t with a stake St, the lower the St+1 (>=St), the (weakly) higher the probability that the coordination at t+1 will succeed. Proposition 5: P(succeed at St+1|succeed at St, St <St+1) >= P(succeed at St+1|succeed at S’t, S’t < St <St+1) >= P(succeed at St+1) Stronger predictions? P(succeed at Sh|GR) >= P(succeed at Sh|BB), P(succeed at Sh|GR) >= P(succeed at Sh|SG) 37

38. Stake Patterns 38

39. Conclusions • Gradualism builds high-level weakest-link coordination: • Starting at a low level • Growing slowly • Externality of coordination building (or collapse) across different social groups • Extensions: • Evidence of belief-based learning: belief elicitation • heterogeneity of gradualism effect: information feedback; payoff (production) function