An equivalence result for (Logit) QRE and Level-k

An equivalence result for (Logit) QRE and Level-k Antonio J Morales (UMA) David Cooper (FSU), Enrique Fatas (UEA), April 15th, 2016 IMEBESS (Rome) Introduction •This paper is part of a long running research programme on boundedly rational models, joint with Enrique and David. –It all started with a “numerical coincidence” for the Minimum Effort Game (EconLett, 2013) –It has continued with “this paper” –It all will end with an paper on an “intuitive contradiction” The numerical coincidence •Fatas and Morales (EconLett, 2013) for the Minimum Effort Game, Goeree and Holt (2005) For the logit predictions, estimation of a noise parameter is required. The objective is to match actual behaviour 𝜋𝜋𝑖𝑖(𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗) = min{𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗}− 𝑐𝑐𝑥𝑥𝑖𝑖 The numerical coincidence •Fatas and Morales (EconLett, 2013) for the Minimum Effort Game, Goeree and Holt (2005) For Step Thinking, no estimation is needed. level 1 play = Level 2 = Level 3 … 𝜋𝜋𝑖𝑖(𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗) = min{𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗}− 𝑐𝑐𝑥𝑥𝑖𝑖 The numerical coincidence •Out of these results, there were two research questions 1. Why is the Logit best estimation so close to level k? 2. Why is actual behaviour so close to predictions? •This paper is about Research Question #1 Was it just a coincidence?? Nash Play = Perfect best response + Correct beliefs Perfect Best Response Noisy Naive Beliefs Correct Nash QRE Level-k NI Step-Thinking •A “simplification” of NE •Models based on best responses without mutual consistent beliefs –Level-k (Nagel, 1995 and Stahl and Wilson, 1994, 1995) –Cognitive Hierarchy (Camerer et al, 2004) •Types of players: –L0 players: Typically choose randomly –L1 type best responds to L0 –Lk types best respond to lower types Quantal Response Equilibrium •A “generalization” of NE •Choices are positively but imperfectly (errors) related to expected payoffs… –QRE (McKelvey and Palfrey, 1995) –Logit version (Anderson, Goeree and Holt, 2002) •Fixed point (equilibrium approach) •is a noise parameter µ The conventional wisdom •Their different nature... –QRE is an equilibrium model that requires “learning” -> Long run –Level-k simplifies others’ decisions-> initial stage •… suggests a differential time horizon •Experimental literature focuses on their differences: –Logit QRE: Holt et al…(Goeree and Holt, AER 2001) –Level-k: Crawford et al (JEL 2014) The Equivalence Result (-) (-) (0) •The most notorious applications of the logit QRE comply with these conditions •In many applications, the logit equation cannot be explicitly solved. But the simpler “level-k equation” can be easily solved Level-k predictions within games with the local payoff property •A“nice” property of level k predictions within the class of games with the local payoff property is that level k predictions do not depend on the distribution of types in the population –In games with continuous strategy space, the optimal behaviour of higher types in these auction-like games will be arbitrarily close the optimal choice of a type-1 player –There is no freedom of degree Pure coordination game (Goeree and Holt, 2005) with n=2 Model Predicted efforts 𝑐𝑐= 0.25 𝑐𝑐= 0.75 LQRE 𝜇𝜇=10 150 129 Step Thinking 155 125 Effort interval [110, 170] Travellers’ Dilemma (Basu, 1994) •Two travellers have lost their baggage, which happen to be identical •The airline announces that they will be reimbursed the lowest of their claims but that a reward Rwill be transferred from the highest to the lowest claimant •Unique NE for all values of R: lower bound of the claim interval (undercutting argument) Capra et al (1999) Model Predicted claims R=5 R=10 R=20 R=25 R=50 R=80 LQRE (𝜇𝜇=10) 181 173 151 136 112 82 Step thinking 190 180 160 150 100 80 Claim interval [80, 200] Imperfect price competition •Two firms •The market price is the minimum price •The low price firm gets her price (the minimum price) •The high price firm gets a fraction α of the minimum price (In Bertrand, α=0) Capra et al, (2002) Model Predicted prices 𝛼𝛼= 0.2 𝛼𝛼= 0.8 LQRE 𝜇𝜇=10 78 127 Step Thinking 88 133 Price interval [60, 160] Why LQRE is so close to level-k? •Two points here: 1. Logit QRE predictions are quite insensitive to the noise parameter (Within the class of games with the local payoff property) within reasonable values –There is a footnote on this in Anderson et al (2002) –Within this class, there is no degree of freedom (remember Haile et al (AER, 2008)) 2. What seems to matter is that some kind of randomness is added to the decision making process, not the logit structure of LQRE Randomness, which randomness?? •The level-k predictions are based on the uniform distribution •Level-k was “conceived” for one shot games (or initial play) while Logit QRE is for later rounds •I have shown data on last periods of play and compare it with level-k Recall the numerical coincidence •Fatas and Morales (EconLett, 2013) for the Minimum Effort Game, Goeree and Holt (2005) For Step Thinking, no estimation is needed. level 1 play = Level 2 = Level 3 … 𝜋𝜋𝑖𝑖(𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗) = min{𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗}− 𝑐𝑐𝑥𝑥𝑖𝑖 Randomness, which randomness?? •When subjects play a game several times, they may learn about two things: 1. About others’ play 2. About the game •Can we replicate the treatment effects within the class of games with the local payoff property in one shot games? –If so, logit beliefs run into problems… New experiments -Experimental design •One-shot games •Discrete strategy space: (110, 120, 130, … 190, 200) •Within subject analysis: Subjects made 20 decisions –They played 5games (imperfect price competition, minimum coordination, travellers’ dilemma, “11-20”game and all-pay auction). –They played 4variations for each game: Player 2 (α2) Player 1 (α1) Low, Low Low, High High, Low High, High LOW: 20 HIGH: 80 New experiments - procedures •Experiments conducted at U. of Valencia •≈ 1½ hours, 18 –20 euros •Four sessions, 224 subjects •All subjects play all 5 classes of games –Minimum game, Travelers’ dilemma and Imperfect price competition rotated across sessions –11 –20 and All-Pay always last two classes •No feedback, subjects paid for one randomly selected game •All games were played with paper & pencil •Initial Instructions (on how to read payoff matrices) •Each class of games was handed out as a separate packet •Each class had a set of instructions, comprehension questions, and four payoff tables. B1-D1 Participante 2 (20%) 110 120 130 140 150 160 170 180 190 200 Participante 1 (20%) 110 66 110 110 110 110 110 110 110 110 110 66 22 22 22 22 22 22 22 22 22 120 22 72 120 120 120 120 120 120 120 120 110 72 24 24 24 24 24 24 24 24 130 22 24 78 130 130 130 130 130 130 130 110 120 78 26 26 26 26 26 26 26 140 22 24 26 84 140 140 140 140 140 140 110 120 130 84 28 28 28 28 28 28 150 22 24 26 28 90 150 150 150 150 150 110 120 130 140 90 30 30 30 30 30 160 22 24 26 28 30 96 160 160 160 160 110 120 130 140 150 96 32 32 32 32 170 22 24 26 28 30 32 102 170 170 170 110 120 130 140 150 160 102 34 34 34 180 22 24 26 28 30 32 34 108 180 180 110 120 130 140 150 160 170 108 36 36 190 22 24 26 28 30 32 34 36 114 190 110 120 130 140 150 160 170 180 114 38 200 22 24 26 28 30 32 34 36 38 120 110 120 130 140 150 160 170 180 190 120 •Our data –remember, one shot game-looks similar to earlier datasets gathered for these games –remember, last periods-. Descriptive results Minimum Game Cooper, Fatas, Morales, and Qi One shot 0% 10% 20% 30% 40% 50% 110 120 130 140 150 160 170 180 190 200 High Cost Low Cost Goeree and Holt (2005) Rounds 8-10 0% 10% 20% 30% 40% 50% 110 120 130 140 150 160 170 High cost Low cost 0% 10% 20% 30% 40% 50% 60% 70% 80% 110 120 130 140 150 160 170 180 190 200 Cooper, Fatas, Morales, and Qi High Reward Low Reward 0% 10% 20% 30% 40% 50% 60% 70% 80% 80 90 100 110 120 130 140 150 160 170 180 190 200 Capra et al, 1999 (Rounds 8 –10) High reward Low reward Travelers’ dilemma 0% 10% 20% 30% 40% 50% 110 120 130 140 150 160 170 180 190 200 Cooper, Fatas, Morales, and Qi Low Alpha High Alpha 0% 10% 20% 30% 40% 50% 60 70 80 90 100 110 120 130 140 150 160 Capra et al Low Alpha High Alpha Imperfect Competition