Introduction Kramnik Pre- vs. De- scriptive Bounded Models Conclusions

Whatever Could I Have Been Thinking?

Peter J. Hammond

February 11, 2010

Experimental and Behavioural Forum, U. of Warwick, 11 February 2010 1/ 24 Introduction Kramnik Pre- vs. De- scriptive Bounded Models Conclusions

Outline

1 Introduction

2 Kramnik

3 Pre- vs. De- scriptive

4 Bounded Models

5 Conclusions

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My Mis-spent (?) Youth

Chess books borrowed from Ipswich (1956–8) and Croydon (1958–66) public libraries. Personal copy of: Gerald Abrahams The Chess Mind (1951) Description of my chess playing: full of mistakes, but sometimes my opponent made even more! Prescription for my chess playing: Play better, or give up! Contrast between descriptive and prescriptive.

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Case Study: (Hu)man versus Machine

Source: http://www.chessbase.com/newsdetail.asp?newsid=3512 Experimental and Behavioural Forum, U. of Warwick, 11 February 2010 4/ 24 Introduction Kramnik Pre- vs. De- scriptive Bounded Models Conclusions

“The blunder of the century” Second game of six, played in Bonn on 27.11.2006. After the move 34. N×f8 (taking a Black rook) . . .

. . . Kramnik (as Black) played 34 . . . Qa7–e3??. To which the response was 35. Qe4–h7 # 1–0. Experimental and Behavioural Forum, U. of Warwick, 11 February 2010 5/ 24 Introduction Kramnik Pre- vs. De- scriptive Bounded Models Conclusions

Etymology

blunder: mid-14c., “to stumble around blindly,” from O.N. blundra “shut one’s eyes”; from (hypothesized) Proto-Indo-European (PIE) base *bhlendh- (see blind). Meaning “make a stupid mistake” is first recorded 1711.

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Body language

Kramnik played the move 34. . . . Qe3 calmly, stood up, picked up his cup and was about to leave the stage to go to his rest room. At least one audio commentator also noticed nothing, while Fritz operator Mathias Feist kept glancing from the board to the screen and back, hardly able to believe that he had input the correct move. Fritz was displaying mate in one, and when Mathias executed it on the board . . .

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Deep Fritz 10’s human interface

. . . Kramnik briefly grasped his forehead, took a seat to sign the score sheet . . . Experimental and Behavioural Forum, U. of Warwick, 11 February 2010 8/ 24 Introduction Kramnik Pre- vs. De- scriptive Bounded Models Conclusions

Press Conference

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Kramnik’s Statement: Model Fixation It was actually not only about the last move. I was calculating this line very long in advance, and then recalculating. It was very strange, some kind of blackout. I was feeling well, I was playing well, I think I [my position] was pretty much better. I calculated the line many, many times, rechecking myself. I already calculated this line when I played 29...Qa7, after each move I was recalculating, again, and again, and finally I blundered mate in one. (This was 5 moves later.) Actually it was the first time that it happened to me, and I cannot really find any explanation. I was not feeling tired, I think I was calculating well during the whole game... It’s just very strange, I cannot explain it.

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Normative Analysis After the move 34. N×f8 (taking a Black rook) . . .

. . . could Black do better than 34. . . . Qe3? Definitely. 34 . . . Kg8 35.Ng6 B×b2 allows White to force perpetual check by 36.Qd5+ Kh7 37.Nf8+ Kh8 38.Ng6+. It is the right way to end the game, as a draw. Experimental and Behavioural Forum, U. of Warwick, 11 February 2010 11/ 24 Introduction Kramnik Pre- vs. De- scriptive Bounded Models Conclusions

Prescriptive decision theory

Standard prescriptive decision theory assumes “unbounded rationality”. It imposes no bounds on the complexity of decision models. It has no way to explain Kramnik’s “blunder of the century”. It would like to rely on preferences revealed by individual behaviour. But individual behaviour generally fails to reveal consistent preferences. And even if it does, the normative significance of those preferences depends on questionable value judgements.

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Descriptive decision theory

Descriptive theory, however, largely focuses uncritically (or unjudgementally) on what people actually do. This makes it almost useless for any normative theory, except one that emphasizes individual autonomy above all else.

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Daniel’s Four “Simple Approaches”

1 Finite memory. But it is not as though Kramnik forgot something. 2 Limited cognitive processing, or level k rationality. But most of the time Kramnik was calculating 5 or more moves ahead. Eventually, though, he overlooked mate in one. 3 Errors in decision-making, with quantal responses. But Kramnik was achieving 100% accuracy in his moves most of the time. Then, on one critical occasion, 0% accuracy. 4 Rules of thumb. How can they be formulated?

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Satisficing Models? Herbert Simon’s “satisficing” theory could have been adequate if he could have made it empirically applicable. Herbert Simon’s better idea: procedural rationality. A rational procedure is to model the decision problem, possibly very imperfectly, and then choose optimally within that model. Two hypotheses, not yet falsified. 1 Decisions are fully rational if the decision problem is really simple. 2 Among those decisions that are seriously contemplated, and relative to the decision model that is being used, the chosen decision is an optimum. If the latter were true, and we could observe what decisions are seriously contemplated, we could infer revealed preferences among those decisions. Experimental and Behavioural Forum, U. of Warwick, 11 February 2010 15/ 24 Introduction Kramnik Pre- vs. De- scriptive Bounded Models Conclusions

Kramnik’s Blunder Revisited Why did Kramnik blunder with 34. . . . Qe3? Not because he had a good model of the game which showed that the move was good enough to win, or even to draw. Kramnik had completely overlooked the winning reply 35. Qh7, with mate in one. His model of the game did not regard 35. Qh7 as a possibility worth serious consideration. He had satisficed in his choice of model, rather than in his choice of move relative to that model. He also became fixated with that model. As the etymology of the word “blunder” suggests, his model of the game left him entirely blind to the possibility of 35. Qh7.

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Directed Cognition

Xavier Gabaix, David Laibson, Guillermo Moloche, Stephen Weinberg (2006) “Costly Information Acquisition: Experimental Analysis of a Boundedly Rational Model” American Economic Review 96: 1043–68. A model of directed cognition, applied to data derived from Mouselab.

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Attention Data

Experiment conducted at Warwick, in collaboration with Stefan Traub (Bremen). Portfolio choice problems similar to: Syngjoo Choi, Raymond Fisman, Douglas Gale, Shachar Kariv (2007a) “Revealing Preferences Graphically: An Old Method Gets a New Tool Kit” American Economic Review, Papers and Proceedings 97: 153–158. — — (2007b) “Consistency and Heterogeneity of Individual Behavior under Uncertainty,” American Economic Review 97: 1921–1938.

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Decision Efficiency

We will construct a measure of decision efficiency, selected as generously as possible, as a normative standard against which decision performance can be measured. How? General idea: Take the several observed choices of each subject, and fit an expected utility model to them, with one risk-aversion parameter.

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Certainty Equivalence

1 For each subject, and for each possible risk aversion parameter value for that subject, each subject’s choices throughout the whole experiment have an overall certainty equivalent — a single number that is higher for better portfolios. 2 Choose a parameter value for each subject which maximizes that subject’s certainty equivalent, so minimizes an associated measure of decision inefficiency. 3 Use that parameter value to find the certainty equivalent of each possible portfolio the subject could have contemplated or chosen.

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Hypotheses Worth Testing?

1 More overall attention devoted to any choice problem leads to a better (more efficient) decision within it? 2 Better options receive more attention? 3 The final choice is well explained by some kind of attention-weighted multinomial logit model? 4 Learning dynamics, etc.?

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Improvements?

Big issue: follow Gabaix et al. in trying to develop a model that explains the search process as well as the final decision. Possible modification of experiment: instead of giving a fixed time before a decision must be made, specify a maximum time that can shrink suddenly, without ever falling below, say, 5 seconds. Also collect data so that we can track the mouse pointer more precisely. And combine with eye-tracking data?

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Conclusions 1) Even with strong value judgements, the gap between prescription and description excludes normative statements based on behaviour that, for example, violates axioms of revealed preference or reduction of compound lotteries. 2) A weaker hypothesis than in prescriptive theory is that agents are unboundedly rational, but only relative to a greatly simplified decision model whose choice is not readily susceptible to normative evaluation. 3) When combined with some extra assumptions, the hypothesis may have testable implications for the results of experiments where one observes something of the attention process, as well as ultimate decisions.

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More Conclusions

4) It also allows a restricted revealed preference hypothesis, applying only to options that receive serious consideration.

Finally, decision trees become enlivened in case the decision maker is forced to recognize the possibility of events which were excluded from earlier decision models. These are true “black swans”, unlike Taleb’s. Enlivened trees were what I talked about in November 2008, so I am going steadily backwards ... Thanks for coming!

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