School of Economics University of East Anglia Norwich NR4 7TJ, United Kingdom

1. School of Economics University of East Anglia Norwich NR4 7TJ, United Kingdom How inductive inferences can be grounded on salience alone: some reflections on the emergence of conventions Robert Sugden September 2008

2. I hope I have something new to say about a classic problem of philosophy – the problem of induction. My starting point is the theory of conventions – a field in which economics and philosophy have been mutually supporting (classic sources: Hume; Schelling; Lewis). Conventions are significant because emergence of conventions seems to require inductive learning; but (unlike in natural science), what is being learned is conventional – i.e. the process of learning constructs the reality that we are learning about.

3. What do I mean by inductive reasoning? Premise: so far, (almost) all Xs have been Y. Conclusion: very probably, future Xs will be Y. Old problem of induction Is this inference ‘really’ rational/ valid? Response: perhaps not, but if it works, why bother? Goodman’s problem of induction What makes ‘Y’ (or ‘X’) projectible? Compare ‘all emeralds have been green’ and ‘all emeralds have been grue’ (= green if observed up to now, blue if observed later). My concern is with Goodman’s problem.

4. How Goodman’s problem relates to conventions Two problems for the analysis of conventions: 1. How do conventions reproduce themselves? 2. How do conventions emerge? Lewis (in Convention, 1969) is mainly concerned with (1). I begin with his analysis of the role of inductive reasoning.

5. Lewis’s theory of conventions (simplified) Consider a large populationof individuals. Recurrently, pairs are drawn from this population and engage in a game-like interaction. Because the population is large, individuals learn about the population in general, not about specific other individuals (think of interactions on the roads). In an interaction, each agent chooses from a set of possible actions. The pair of actions chosen is (a1, a2). (a1, a2) is a putative convention if: (i) each agent is motivated to perform his component, conditional on the belief that the other will perform hers; (ii) there is at least one other (a’1, a’2) for which (i) holds; (iii) ‘coincidence of interest predominates’. If almost everyone conforms to such an (a1, a2), this is a convention.

6. Lewis asks: how do conventions reproduce themselves? This reduces to: how are ‘concordant mutual expectations’ sustained? Lewis begins with novel(one-shot) coordination problems (Schelling games), and argues (following Schelling) that mutual expectations converge on salient equilibria. Lewis’s explanation of this: a tendency for people to pick the salient as a ‘last resort’, when they have ‘no strong reason to do otherwise’. In recurrent coordination problems, mutual expectations converge on precedent. But precedence is just another form of salience, i.e. people tend to repeat previously successful actions as a last resort.

7. But Goodman’s problem: any regularity observed in previous interactions can be described in many different ways. How do people converge on the same description? If each interaction is slightly different from all previous ones, how do people converge on the sameconcepts of similarity? Lewis’s answer: ‘we happen uniformly to notice some analogies and ignore others – those we call “natural” or “artificial” respectively ... and fortunately we have learned that all of us will mostly notice the same analogies’. So, each individual treats regularities as projectible if he perceives then as salient. In learning about conventions, this principle works because perceptions of salience are shared, and because of feedback mechanisms. But would the same principle work for learning about non-conventional properties of the world?

8. Second problem: how do conventions emerge? Consider a case in which coordination requires the two players to choose different actions. E.g. Lewis’s example of the cut-off phone call: one individual has to call back, the other has to wait. Usual symmetry-breaking analysis (e.g. Sugden, Skyrms): If players do not recognise any asymmetry of roles, there is a stable mixed-strategy equilibrium in which ‘wait’ and ‘call back’ have equal expected payoff. But if players do recognise an asymmetry, experience-based learning will lead to some asymmetric convention (e.g. ‘caller calls back, called waits’. Why? Because as soon as there is some asymmetric play, everyone benefits by conforming to that asymmetry – so a self-reinforcing tendency towards a convention. But...

9. Problem: for symmetry to be broken, random variation in actual behaviour has to create an accidental regularity (e.g. callers call back more frequently than called) ... ... and individuals have to treat this as projectible. A convention starts to evolve when random variation happens to map on to some pre-existing conception of salience. It seems that people have to use Lewis’s principle of projecting salient regularities even when regularities are accidental.

10. Lewis’s presumptive principle: if you observe a regularity which you perceive as salient, treat it as projectible. I want to argue that this principle is pragmatically defensible – not just for conventions, but for learning in general.

11. Goodman’s ‘third sons’ example Goodman (1954) casts his problem of induction as: ‘What hypotheses are confirmed by their positive instances?’ ‘That a given piece of copper conducts electricity increases the credibility of statements asserting that all copper conducts electricity. But the fact that a given man now in this room is a third son does not increase the credibility of statements asserting that other men now in this room are third sons, and so does not confirm the hypothesis that all men now in this room are third sons ... Plainly, then, we must look for a way of distinguishing lawlike statements [e.g. all copper conducts electricity] from accidental statements [e.g. all men in the room are third sons].’ I disagree!

12. Suppose I am a woman at Goodman’s lecture. There are 50 men in the audience. I start asking them about their positions in birth order of their mothers’ sons. The first man is a third son. (Prior probability  1/14.) Nothing surprising about this, no reason to expect others to be third sons. But what if second, third, ... fifth men are all third sons? This is an extraordinary coincidence. My observations clearly fit a salient pattern, whether accidental or not. Now what if sixth, ..., tenth men are all third sons? Even if I discount the first five (since it was only after them that the ‘third son’ pattern became salient to me), the probability of 5 third sons in a row is about 1/500,000. I infer: this pattern is not accidental. Or: something is going on here.

13. My mode of reasoning: the Salience Schema: I have observed regularity R in domain D. I perceive R as salient. If events in D were governed by chance, salient regularities as extreme as R would occur very infrequently. ___________________________________ Very probably [subjective], R is not an accident. Notice that, to derive the conclusion, R has to be salient, and it must be very unlikely that any salient regularity would occur by chance. This reasoning does not depend on a prior belief/ hypothesis that R will occur. It is activated by surprise.

14. So, it is not an objection that projectibility is being determined by personal conceptions of salience. The individual is not expecting to find his personally salient patterns in the world. He is surprised to find them – surprised because so few of the patterns that the world could throw up are salient to him. And he is qualified to judge that! I claim that progress in natural and social sciences often comes from the observation of highly salient, initially surprising patterns – e.g. the ‘social gradient’ of mortality rates in the British civil service (Marmot), correspondences between sedimentary rocks in Falkland Islands and South Africa.

15. Conclusion: Inductive inferences can be grounded on salience alone.