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Clock Games: Theory and Experiments

John Morgan UC Berkeley. Clock Games: Theory and Experiments. Markus K. Brunnermeier Princeton University. Timing …. Common features of many timing problems time has to be “ripe” Congestion effect: there is only room for K players Waiting motive: first movers risk more

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Clock Games: Theory and Experiments

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  1. John MorganUC Berkeley Clock Games: Theory and Experiments • Markus K. Brunnermeier Princeton University

  2. Timing … • Common features of many timing problems • time has to be “ripe” • Congestion effect: there is only room for K players • Waiting motive: first movers risk more • Uncertainty about rivals’ moves • Examples • Launching a new product • Leading a revolution • Participating in a currency attack • …

  3. … Timing • Differences • Few key players – few cohorts – many players • Rivals’ moves are difficult/easy to predict • Rivals’ moves are observable/unobservable • Observations • Initial delay • Sudden onset of action • Objectives of paper • Provide tractable model • Experimentally verify predictions given complexity of the game

  4. Some related literature • Theory on timing games • Pre-emption games • War of attrition games • Recent papers • Park & Smith (2003) • Morris (1995) • AB (2002,2003) • Experimental literature • McKelvey and Palfrey (1992)

  5. introduction theory model setup unobservable actions observable actions Information clustering experiment conclusion

  6. Model setup sequential awareness (random t0 with F(t0) = 1 - exp{-t0}). exit payoff (random) ‘end of game payoff’ 1 1/ t t0+  t0 t0+  0 random starting point for sure all players are aware of t0 exogenous end of game maximum life-span

  7. Payoff structure • Payoffs • ‘exit payoff’ (random) for first K players • ‘end of game payoff’ for last I-K players • Tie-breaking rule if Kth, K+1th, …, K+nth player exit at the same time t > t0, exiting players receive the exit payoff with equal probability.

  8. introduction theory model setup unobservable actions observable actions information clustering experiment conclusion

  9. Delay - unobserved actions … At t = ti +  benefit of exiting= size of payoff drop benefit of waiting For ! 0 Solving for  ‘end of game payoff’ ti t0

  10. player tj player ti tj ti ti -  since ti  t0 +  since ti  t0 Equilibrium hazard rate – unobserved actions • If everybody waits for  periods, then at ti + Prob(payoff drop at ti +  + ) = =Prob(Kth of others received signal before ti + ) • Random for two reasons: • t0 is random • Timing of Kth signal within window of awareness is random • Condition on fact that payoff drop did not occur

  11. … Delay – unobservable actions Proposition 1: In unique symmetric equilibrium delay .. where is a Kummer function. Integral representation

  12. 140 20  80 120 Delay increases with window of awareness Proposition 2: Delay increases with window of . Why? Makes it more difficult to predict moves of others.

  13. introduction theory model setup unobservable actions observable actions information clustering experiment conclusion

  14. Herding – observable actions Proposition 3: All players exit after observing the first player exiting (if first player exits in equilibrium after receiving the signal). Intuition: backwards induction

  15. Delay of first player – observable action = hazard rate of the first player exiting = probability of not receiving the high exit payoff when herding after first Simplifies to a ratio of Kummer functions

  16. introduction theory model setup unobservable actions observable actions Information clustering experiment conclusion

  17. Information Clustering • Our model - CC model - AB model 1/ t0 t0+  t0 t0+  t0 t0+  I players continuum of playersin I groups continuum of playersno information clustering

  18. t0 t0+  Comparison to AB (unobservable) • Proposition 6: > AB. • 2 effects: • Individual player carries more weight (focus of CC-model) • Synchronization is more complicated • In AB: hazard rate  prob. of being “Kth” player conditional on knowing to “Kth” player, ti knows that next player exits an instant later with probability 1 and causes payoff drop. • In BM: hazard rate  prob. of being “Kth” * prob. K+1th follows in next instant. • Proposition 7: Fix K=I. As I!1, !AB. • (Kummer functions converge to exponentials.)

  19. Comparison with AB (observable) • Proposition 9:1,AB = 0. • Intuition • If at ti + AB,1 payoff hadn’t occurred it will occur with prob. one in next instant (i.e. hazard rate=1) • Hazard rate is continuous • For any AB,1>0 player i has incentive to exit earlier. • Hence, AB,1=0. • Corollary:1 > 1,AB = 0. • Same reasoning as in case with unobservable actions.

  20. Isolating information clustering (CC-model) • Continuum of players, but I cohorts • Difference: player i knows that his cohort exits at ti +  • Before ti+: drop if Kth cohort out of (I-1) exits • After ti+: drop if (K-1)th out of (I-1) exits(since own cohort exited) • At ti+: drop occurs with strictly positive prob.

  21. Isolating information clustering (CC-model) Expected marginal costK-1 out of I-1 Expected marginal costK out of I-1 Marginal benefit g  of i’s cohorts

  22. Isolating information clustering (CC-model) Expected marginal costK-1 out of I-1 Expected marginal costK out of I-1 Marginal benefit g  BM AB Multiple equilibria Reasoning for 1 in case with observable actions is analogous.

  23. introduction theory experiment design and procedures measures – delay & herding results further insights conclusion

  24. 1 t 0 Experimental design … Stock market illustration g=2%, =1%, (½ second) 2 parallel rounds (randomly matched) 6 players per round First 3 sell at exit price egt, others egt0 price “true value”

  25. …Experimental design…

  26. … Experimental design …

  27. … Experimental design …

  28. … Experimental design • Average payout: ECU 30.32 = $15.16 • Hovering to avoid coordination via mouse-click • “Learning by doing:” focus on periods 20,…,45 • Obvious mistakes: sale within 10 periods (5 sec.) • 16 Sessions • Treatments:

  29. introduction theory experiment design and procedure measures – delay & herding results further insights conclusion

  30. Measures • Delay measures • delay texit i – ti • bubble length texit [K] – t0 • Notice censoring! • Herding measure • GAP2,1texit [2] – texit [1] • GAP3,2texit [3] – texit [2]

  31. introduction theory experiment design and procedure measures – delay & herding results further insights conclusion

  32. Theory Predictions *For the Observable treatment, Delay is only meaningful for first seller

  33. Descriptive Statistic

  34. Histograms – Bubble Length Observable Baseline 0.15 113 113 Periods Periods Compressed 0.15 113 Periods

  35. Histogram - Third Player’s Delay Compressed Baseline 0.267 74 74 Periods Periods

  36. Histogram - Gap Length Baseline Observable 0.343 Periods 49 Periods 49 Compressed 0.343 49 Periods

  37. Results – Session Level Analysis • Prediction 1: Bubble Length Baseline > Compressed 5 % • Prediction 2: Delay • Player 1: Baseline > Compressed 5 % • Player 2: 5 % • Player 3: 1 % • Player 1: Baseline > Observable • Prediction 3: GAP • GAP21: Baseline > Observable 5 % • GAP32: 5 % failed to reject =

  38. Results: Delay – Individual Level Analysis

  39. Results: Herding – Individual Level Analysis

  40. introduction theory experiment design and procedure measures – delay & herding results further insights conclusion

  41. t0–effect & exiting before ti • t0-effect • Risk aversion - stakes are higher for large t0 • Difference in risk aversion among players • Delay of first seller < Delay of third seller • Effect becomes larger for large t0 • Misperception of constant arrival rate • Waiting for a fixed (absolute) price increase • Exiting before ti • mistakes • Worries that others suffer t0-effect (risk aversion) • Effect is larger in Baseline since bubble is larger

  42. +0.3 % -7.1 % base rate 8.8 % Probit of Non-Delay

  43. Conclusion • Many timing games have in common • Time has to be “ripe” • Congestion effect • Costly to be pioneer • Uncertainty about others moves • Theoretical predictions of clock games: • Delay increases with • number of key players • uncertainty about others moves • Herding/sudden onset if moves are observable • Initial delay for first player decreases with number of players • Experiment • Comparative static/Treatment effects are confirmed • Delay and herding less strong (in terms of levels) • Additional insights: t0-effect, …

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