Guessing with negative feedback : an experiment Angela Sutan, Marc Willinger ESC & LAMETA Dijon - Montpellier Lorentz workshop, October 2007
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
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
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
Assumptions about eductive reasoning in BCG Agents choose undominated strategies Iterated elimination of dominated strategies Common knowledge of rationality
Laboratory experiments : Nagel (1995) p Median Mean ½ 17 27,05 2/3 33 36,73 4/3 66 60,12 Equilibrium (0) never observed 6% of subjects choose n < 10 Average depth of reasoning 2 3
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)
Newspapers experiments : Nagel et al. (2002) P=2/3 Peaks at 33, 22 and 0 Depth of reasoning : 1, 2 infinity
Positive feedback : monotonic convergence High mean high winning number Low mean low winning number Expected mean Depth of reasoning
Negative feedback : non-monotonic (oscillatory) convergence High mean low winning number Low mean high winning number Expectation Depth of reasoning
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)
Negative feedback Iterated elimination of dominant strategies b 0 1 b 3 E b 4 b 2 100
Equilibrium : E = 100 1 ( 1) n 1 + p p n + ( 1) n p n 100 n (with p < 1) E 100 = 1 + p
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
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
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
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)
Positive feedback interior equilibrium experiment (Gueth et al., 2002) 1/2 (mean + 50) Starting at 100 : 75-62.5-56.25-53.125 50 Starting at 0 : 25-37.5-43.75-46.975 50 Results : Very fast convergence to 50 compared to corner solution High frequency of equilibrium chosen (almost 50%) But! 50 is a focal point
Our experiment 100 p mean (p < 1) Single round experiment Multiple-rounds experiment
Single-round experiment Group size : 8 subjets Rule E Average Abs groups deviation subjects BCG+ 2/3 (mean + 30) 60 50.0 9.9 11 88 BCG- 100 2/3 mean 60 60.4 5.1 12 96 Significantly lower abs deviation for BCG- (Wilcoxon-Mann-Withney, 5%)
76 80 84 88 92 96 100 76 80 84 88 92 96 100 72 72 68 68 64 64 35 30 25 20 15 10 5 0 single-round BCG- (p = 2/3) 36 40 44 48 52 56 60 single-round BCG+ (p = 2/3) 36 40 44 48 52 56 60 32 32 28 28 24 24 20 20 16 16 12 12 8 8 4 4 0 0 18 16 14 12 10 8 6 4 2 0
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 : 100 - p mean 2 treatments : p = 2/3 E = 60, p = ¼ E = 80 Max gain per period : 8 Euros. Max gain observed : 32 Euros
Data p=2/3 number of choices 50 40 30 20 10 0 30 40 50 60 70 80 per1 per6
Data p=1/4 number of choices 50 40 30 20 10 0 30 40 50 60 70 80 90 100 per1 per6
Winning numbers, p = 2/3, 9 groups 80 70 60 50 40 0 1 2 3 4 5 6 7 8 9 10
Winning numbers, p=1/4, 7 groups 100 95 90 85 80 75 70 0 1 2 3 4 5 6 7 8 9 10
Average numbers for BCG+ (p =2/3)
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 ),
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?
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-?
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
Boundaries boundaries of succesive strategy intervalls 120 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 depth of reasoning BCG+ BCG-
size of remaining intervall (p = 2/3) 120 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 BCG- BCG+
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)
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 0 0 100 E Negative feedback b 1 b 0 0 E 100
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 0 0 100 E Negative feedback b 1 b 2 b 0 0 E 100
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
p = 2/3 0,6 0,5 0,4 0,3 0,2 0,1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 BCG- BCG+ p = 1/4 0,6 0,5 0,4 0,3 0,2 0,1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 BCG- BCG+
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
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