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Strategic Timing Games and Decision Delays: Models, Analysis, and Experiments

This research paper delves into the strategic aspects of timing games, exploring the impacts of delay, uncertainty, and rival actions. The study provides a theoretical framework, experimental validation, and comparisons with existing models. It addresses various scenarios like product launches, revolutions, and currency attacks to derive key insights. The paper also examines information clustering in decision-making and the strategic implications of observable versus unobservable actions.

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Strategic Timing Games and Decision Delays: Models, Analysis, 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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