Unit IV: Thinking about Thinking

Unit IV: Thinking about Thinking

•Choice and Consequence•Fair Play•Learning to Cooperate•Summary and Conclusions

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Learning to Cooperate

• The Bar Problem• The Logic of Collective Action• Changing the Rules of the Game• The Problem of Trust• Limits of Strategic Rationality• Tournament Update

The standard, deductive model of decision-making break down for two reasons:

(i) Bounded rationality: human decision making is limited by finite memory and computational resources.

(ii) Recursiveness: thinking about others’ thinking involves forming subjective beliefs, and subjective beliefs about subjective beliefs, and so on.

Last Time

Deductive models describe actual human behavior only in very simple decision contexts. In more complicated environments, the computational requirements to deduce a solution quickly swamp the capacity of any human reasoning. Chess appears to be well beyond the ability of humans to fulfill the requirements of traditional deductive reasoning.

In today’s “fast” economy a more dynamic theory is needed. The long-run position of the economy may be affected by our predictions!

“On Learning and Adaptation in the Economy,” Arthur, 1992, p. 5

Last Time

[H]umans, in economic decision contexts that are complicated or ill-defined … use not deductive, but inductive reasoning. That is, in such contexts we induce a variety of working hypotheses or mental models, act upon the most credible, and replace hypotheses with new ones if they cease to work. Inductive reasoning leads to a rich psychological world in which an agent's hypotheses or mental models compete for survival against each other, in an environment formed by other agents' hypotheses or mental models -- a world that is both evolutionary and complex. Inductive reasoning can be modeled in a variety of ways.

“Inductive Reasoning & Bounded Rationality,” Arthur, AER, 1994

The Bar Problem

The Bar Problem

(Arthur, 1994)

Each week, a group of people decide whether or not to go a particular bar. Space is limited, so when too many people go, no one enjoys the experience.

There is no way to know for sure how many will attend, but a person will

• “go” if she expects less than X show up• “stay home” if she expects more than X to go

The Bar Problem

There is no obvious (deductive) model to predict attendance.

Common expectations are self-defeating

• If all believe few will go, all will go• If all believe many will go, none will go

“Oh that place is so crowded, nobody goes there any more” – Yogi Berra

“Oh that place is so crowded, nobody goes there any more”

– Yogi Berra

The Bar Problem

Let’s assume there are 100 people who might go to the bar on particular night, and an individual will go if she expects less than 60 to attend, stay home, if she expects 60 or more.

Each individual can form several predictions as a function of past attendance. For example, attendance over the past w weeks might be:

…44 78 56 15 23 67 84 34 45 76 40 56 22 35

The Bar Problem

…44 78 56 15 23 67 84 34 45 76 40 56 22 35

They may have predictors such as: predict next week’s attendance to be

– the same as last week’s [35]– a mirror image around 50 of last’s week [65]– 67 [67]– a (rounded) average of last 4 weeks [49]– the trend in last 8 weeks [29]– the same as 2 weeks ago (2-period cycle detector) [22]– the same as 5 weeks ago (5-period cycle detector) [76]– etc….

The Bar Problem

…44 78 56 15 23 67 84 34 45 76 40 56 22 35

Each agent can only keep track of a subset of possible predictors. She decides to go or stay home based on the most accurate predictor in her current set (the active predictor). After all decisions are made, each agent learns the new attendance and updates her active predictors.

The process might look like this:

The Bar Problem

100 90 80 70 60 50 40 30 20 10 0

0 20 40 60 80 100 Weeks

Source: Arthur, 1994

AttendanceOn average, 60 people

attend each week.

The Bar Problem

“The predictors self-organize into an equilibrium pattern or “ecology” in which of the active predictors, those most accurate and therefore acted upon, on average 40% are forecasting above 60, 60% below 60.”

“[I]f we view it as a pure game of predicting, a mixed strategy of forecasting above 60 with probability 0.4 and below [60] with probability 0.6 is Nash”

- Arthur, 1994, p. 8.

The Logic of Collective Action

“[T]he achievements of a union, even if they were more impressive than the staunchest unionist claims, could offer the rational worker no incentive to join; his individual efforts would have no noticeable effect on the outcome, and whether he supported the union or not he would still get the benefit of its achievement.”

- Olson, 1965, p. 76.

“Given a Prisoner’s Dilemma such as trade union formation, it can be argued that prolonged interaction between the workers will make the real payoff structure diverge from the purely monetary one, because the welfare of others will enter into the utility function of the individual. This could change the game into an Assurance Game […] sustained interaction in such communities may do away with the problem itself by transforming the nature of the game.”

- Elster 1979, p.146.



Prisoner’s Dilemma Assurance Game

6, 6 0, 5

5, 0 1, 1

In the PD, the pareto efficient outcome is individuallyinaccessible and collectively unstable; in the AG, it is individuallyinaccessible but collectively stable, or self-enforcing.

3, 3 0, 5

5, 0 1, 1

Repeated interactions provide the conditions necessary for cooperation by transforming the nature of the game in two ways:

“Enlarge the shadow of the future”

Increase the amount of information in the system. This may reduces

strategic uncertainty () and allow players to coordinate their expectations and behavior on mutually beneficial outcomes.

1

0

T-RT-P


The Folk Theorem

(R,R)

(T,S)

(S,T)

(P,P)

The shaded area is the set of SPNE. The segment PP,RR is the set of “collectively stable” strategies,for ( > *).



Repeated interaction is a necessary but not a sufficient condition for cooperation.

To cooperate, those involved must regard the outcome as fair, and this may be subverted by strategic attempts to manipulate the process.

From this point of view, trust can be seen as the problem to be solved.

A thought experiment. Workers in a young industry consider the option of forming a union. A meeting is held and a vote is called. A majority oppose. A worker known to have been an instigator stands and speaks: “Brothers and sisters, do not let your petty differences stand in the way of our purpose. Workers – all of us – are called upon to unite against a common enemy: the bosses who exploit us and grow fat off our divisiveness. Together we possess a might they cannot ignore; alone we fight – one against another – for the scraps that fall from their tables.”


the Common Enemy


“No union can function for a day in the absence of some rudimentary notions held by the members that being a member is of value in itself, that the individual organization costs … have to be accepted as necessary sacrifices, and that each member is legitimately required to practise solidarity and discipline….”

(Offe and Wiesenthal, 1980, p. 184).

Two s


“The logic of collective action of the relatively powerless differs from that of the relatively powerful in that the former implies a paradox that is absent from the latter - the paradox that interests can only be met to the extent they are partly redefined.”

(Offe and Wiesenthal, 1980, p. 184).

Two s

Changing the Rules of the Game

On the one hand, a game of strategy is defined by its rules, and the assumption of strategic rationality supposes that each player is following preferences and decision procedures that he has determined for himself -- that is, monologically, regardless of whether or not he agrees therein with the other players. In non-zero sum games there are mutual gains to be had if a bargain is achieved, but the entire process takes place within a set of predetermined and irreducibly opposed interests, or preferences.

On the other hand, changing the rules of the game requires that the players first try to understand each other, and in so doing come to understand their own positions in a new light, reflected in the accounts and descriptions offered by others. Instead of bargaining from the relative strength of their interest positions and achieving a compromise, circumscribed by their given positions, they may even redefine their preferences.


From this point of view, trust can be seen as the problem to be solved. It is no simple matter to provide reasons for each other’s trust – why one should be believed to mean what is said and intend what is proposed – but to change the rules of the game, those involved must regard the terms of their agreement as fair, as mutually binding, and this may be subverted by strategic attempts to manipulate the process.


The Problem of Trust

• is a “social lubricant” (Arrow, 1974, p. 23). It’s easier to make efficient exchanges (transactions costs are lower) when trust is higher.

• is a reciprocal expectation; it’s hard to trust other(s) without thinking yourself trustworthy. So is the absence of trust; I may fear you will betray me not because I believe you are untrustworthy, but because I know I am.

• may confer an evolutionary advantage on groups.

• is a state that cannot be willed.

Trust

Rational analysis, for all its inadequacies, is indeed the best instrument of cognition we have. But it is often at its best when it reveals to us the nature of the situation we find ourselves in, even though it may have nothing to tell us how we ought to behave in this situation. Too much depends upon our choice of values, criteria, notions of what is “rational,” and, last by but no means least, the sort of relationship and communication we establish with the other parties in the “game.” (Rapoport, 1960, p. 214).

The Limits of Strategic Rationality

These choices have nothing to do with the particular game we are playing. They are not strategic choices, i.e., choices rationalized in terms of the benefits they bestow on us in a particular conflict. Rather they are choices which we make because of the way we view ourselves, and the world, including the other players. The great philosophical value of game theory is its ability to reveal its own incompleteness. Game-theoretical analysis, if pursued to its completion, perforce leads us to consider other than strategic modes of thought (Rapoport, 1960, p. 214).

The Limits of Strategic Rationality

Preliminary Tournament ResultsBaseline

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Unit IV: Thinking about Thinking