Guessing with negative feedback : an experiment. Angela Sutan, Marc Willinger ESC & LAMETA Dijon - Montpellier

Transcription

1 Guessing with negative feedback : an experiment Angela Sutan, Marc Willinger ESC & LAMETA Dijon - Montpellier Lorentz workshop, October 2007

2 Depth of reasoning in a strategic environment Cognitive process of guessing Game theory : strategic reasoning Psychology : Machiavellic Intelligence (Theory of Mind) : strategic manipulation, Depth of reasoning = number of reasoning steps Not observable Based on the assumption of common knowledge of rationality

3 The beauty contest game with positive feedback (Moulin, 1986) M players choose simultaneously a real number in the closed interval [0, 100] The winner is the player whose chosen number is closest to : p mean Iterated elimination of dominated strategies leads to : Equilibrium = 0 If p < 1, the equilibrium is eductively stable If p = 1 infinite number of equilibrium solutions, p > 1 two corner solutions

4 Example : p = 0,7(Ho,Camerer,1998) Choosing a number in the interval : [70, 100] depth of reasoning 0 [49, 70[ depth of reasoning 1 [34, 49[ depth of reasoning 2

5 Assumptions about eductive reasoning in BCG Agents choose undominated strategies Iterated elimination of dominated strategies Common knowledge of rationality

6 Laboratory experiments : Nagel (1995) p Median Mean ½ 17 27,05 2/ ,73 4/ ,12 Equilibrium (0) never observed 6% of subjects choose n < 10 Average depth of reasoning 2 3

7 Most likely type of reasoning : iterated best-reply (to lower depth of reasoning) Level 0 : choose randomly 50 Level 1 : best reply to level 0 p 50 Level 2 : best reply to level 1 p(p 50)

8 Newspapers experiments : Nagel et al. (2002) P=2/3 Peaks at 33, 22 and 0 Depth of reasoning : 1, 2 infinity

9 Positive feedback : monotonic convergence High mean high winning number Low mean low winning number Expected mean Depth of reasoning

10 Negative feedback : non-monotonic (oscillatory) convergence High mean low winning number Low mean high winning number Expectation Depth of reasoning

11 Beauty contest game with negative feedback M players choose a number in the interval [0,100]. The winning number is : 100 p mean (p < 1)

12 Negative feedback Iterated elimination of dominant strategies b 0 1 b 3 E b 4 b 2 100

13 Equilibrium : E = ( 1) n 1 + p p n + ( 1) n p n 100 n (with p < 1) E 100 = 1 + p

14 Why should negative feedback matter? Many real markets provide negative feedback on expectations : crop producers : a high price expectation will lead to low market price financial markets : a low price expectation will lead to a high market price Alternating correction of expectation errors (reducing the bias on both sides of the equilibrium point) facilitates convergence BCG experiments with interior equilibrium exhibit smaller deviations even in first round choices In repeated games, negative feedback reduces the anchoring bias on previously observed values Negative feedback acts as a stabilizer : a change in one direction will be offset by the reactions to move in the opposite direction. Heemeijer, Hommes, Sonnemans, Tuinstra (2007) : Negative feedback market (Cobweb) : Prices are stable and often converge to the equilibrium Positive feedback market (CAPM) : Prices fluctuate and often deviate from fundamentals

15 Eductive reasoning in the Cob-web model P Supply Price p 0 p 2 p 4 p 3 p 1 Demand x 1 x 3 x 2 x 4 x 0 x Quantity

16 Eductive reasoning in the Cob-web model P p 0 p 2 p 4 Supply p 3 p 1 Demand x 1 x 3 x 2 x 4 x 0 x

17 BCG with interior equilibrium M players choose numbers in the intervall [0, 100]. BCG+ : the winning number is : p (mean + c) (p < 1) BCG- : the winning number is : 100 p mean (p < 1) E-stable equilibrium : BCG+ : cp/(1-p) BCG- : 100/(1+p)

18 Positive feedback interior equilibrium experiment (Gueth et al., 2002) 1/2 (mean + 50) Starting at 100 : Starting at 0 : Results : Very fast convergence to 50 compared to corner solution High frequency of equilibrium chosen (almost 50%) But! 50 is a focal point

19 Our experiment 100 p mean (p < 1) Single round experiment Multiple-rounds experiment

20 Single-round experiment Group size : 8 subjets Rule E Average Abs groups deviation subjects BCG+ 2/3 (mean + 30) BCG /3 mean Significantly lower abs deviation for BCG- (Wilcoxon-Mann-Withney, 5%)

21 single-round BCG- (p = 2/3) single-round BCG+ (p = 2/3)

22 Multiple-rounds experiment Computerized (z-tree). 128 subjects Partner design : 8 subjects per group 10 rounds Average time = 40 min Strategy space : real number between 0 et 100. Winning number : closest to : p mean 2 treatments : p = 2/3 E = 60, p = ¼ E = 80 Max gain per period : 8 Euros. Max gain observed : 32 Euros

23 Data p=2/3 number of choices per1 per6

24 Data p=1/4 number of choices per1 per6

25 Winning numbers, p = 2/3, 9 groups

26 Winning numbers, p=1/4, 7 groups

27 Average numbers for BCG+ (p =2/3)

28 Depth of reasoning : Cognitive hierarchy model (Camerer et al. 2004) Main assumption : heterogeneous depth of reasoning An agent of level k believes that other players are of a lower level (l < k) The number of players of level k is decreasing with k (Poisson law) : f τ k ( k ) = e τ / k! Relative frequency of level h players for a level k player g k ( h ) = f ( h ) / k l = 0 f ( l ),

29 Estimated τ : p = 2/3, E = 60 : τ - = 1,55 ( observed mean = 56,46) p = ¼, E = 80 : τ - = 0,94 (observed mean = 78,04)) Positive feedback : 1 < τ + < 2 Why such a difference between the p = 2/3 and p = 1/4?

30 Main findings for negative feedback Winning numbers are close to the predicted equilibrium number (even in period 1) Winning numbers converge to the equilibrium number after a few rounds (mostly from above) The average estimated depth or reasoning is between 1 and 2 Puzzle : Since the average depth of reasoning is equal in BCG+ and BCG-, why are winning numbers closer to the equilibrium number in BCG-?

31 Hypothesis Subjects trade off expected information vs cost of mental effort 1. Subjects have limited cognitive ability (cognitive constraint) cost of mental effort 2. The marginal cost of mental effort is increasing, and becomes infinite beyond the cognitive constraint 3. An additional step of reasoning is carried out only if the (expected) marginal benefit is larger than the marginal cost of reasoning 4. The benefit of a additional step of reasoning is a function of the expected marginal information

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33 Boundaries boundaries of succesive strategy intervalls depth of reasoning BCG+ BCG-

34 size of remaining intervall (p = 2/3) BCG- BCG+

35 Marginal benefit of expected information At each step, numbers between the previous boundary and the new located boundary are mentally scanned. The localization process implies that the equilibrium point is identified with increasing precision after each step Useful information gained after each step is measured by the intersection between scanned numbers and the nondominated strategies intervall. SNARC effect (Dehaene, 1993) : numbers are perceived from left to right on a logarithmic ordinal scale (for left to right writing cultures)

36 Available information : number scanned between two successive bounds (SNARC effect) Useful information : available information dominant strategy intervall Positive feedback (corner solution) b 2 b 1 b E Negative feedback b 1 b 0 0 E 100

37 Available information : number scanned between two successive bounds (SNARC effect) Useful information : available information dominant strategy intervall Positive feedback (corner solution) b 2 b 1 b E Negative feedback b 1 b 2 b 0 0 E 100

38 Marginal information H ( I ) = p log2 p Marginal information at stage k : H b b k 1 k k 1 ( Ik ) = log2 h l b b h l k + H ( I k ) = 0,01 Log 2 0,01

39 p = 2/3 0,6 0,5 0,4 0,3 0,2 0, BCG- BCG+ p = 1/4 0,6 0,5 0,4 0,3 0,2 0, BCG- BCG+

40 In early steps marginal information is always larger in BCG- than in BCG+ For any increasing cost of reasoning function, the number of steps is at least as large for p = 2/3 than for p = ¼

41 Conclusions Chosen numbers are close to E under negative feedback Convergence is faster than under positive feedback The cognitive hierarchy model predicts that the average depth or reasoning is the same under negative than positive feedback Early steps are more informative under negative feedback