Game Theory and Rationality

John Milanese

Rational Choice Theory

Seeks to model and derive claims about what it means to make decisions rationally.Assumptions:

(1) “Utility” is a unit to keep track of preferences for goods or alternatives. (2) Agents have complete rankings of the alternatives they face: for any two alternatives x and y, either the agent ranks x above y (prefers x to y), or the agent prefers y to x, or the agent is indifferent. (3) These preferences for alternatives are transitive.(4) More of a good is always preferable.

• For any two goods, there is a quantity at which the two goods could be substituted without change to utility.

• The relationship relating the rates at which goods are substitutable without change in utility is called an indifference curve

Game Theory• A subset of rational choice theory, where we care about

modeling rational choice under strategic interaction:– Information about the choice situation (or game) is complete and

symmetric (like centipede). (these assumptions can be relaxed).– I know how much utility is associated with my choices given you

might choose to do, and I know what your utility levels could be too, and you know what mine is, and you know that I know what yours is, I know that you know what mine is, and you know all this, and I know you know all this….

– Communication is not allowed (this can be relaxed too – “signaling” and “threats” enter the analysis).

– Outcomes of simultaneous choices can be represented on a grid (a payoff matrix), and sequential games are represented on a tree.

Payoff Matrix for a Simple Game

(4,4) (1,2)

(2,1) (0,0)

Player 2

Player 1

C1

C2

C1 C2

What is the rational choice to make?

• Suppose you are player 1. – If player 2 chooses C1, then you ought to pick C1.– If player 2 chooses C2, then you ought to pick C1.Therefore, you ought to pick C1 no matter what.

C1 is a “dominant strategy”: C1 maximizes utility no matter what others do. No choice is better (others may be equal), and so C1 is the rational choice.

The point:

If there is a dominant strategy, then this seems like a clear case of what action would be the rational choice.

We’ll worry about this in a few minutes…

The“Stag Hunt Game”

• You’re player 1, hunting a deer with player 2. In order to catch the deer, both players must choose C1. But rabbits run through the woods. Each of you could be distracted and choose to just catch a rabbit by yourself instead (C2).

• What’s rational?

Payoff Matrix for Stag Hunt

(4,4) (1,2)

(2,1) (2,2)

Player 2

Player 1

C1

C2

C1 C2

What is the rational choice to make?

There is no dominant choice or strategy. But there are Nash Equilibriums:

A pair of choices (or strategies) is a Nash Equilibrium when neither player has a unilateral incentive to deviate from that strategy pair (each choice is a rational response to the other).

<C1,C1> is an equilibrium, and so is <C2,C2>…

• In general, all dominant pairs of choices are Nash Equilibriums (like the 1st game). But Nash Equilibrium strategies are not always dominant choices (Stag Hare Game).

• Notice that C1 is a rational choice: the <C1,C1> equilibrium “dominates” the <C2,C2> equilibrium. Player 1 should be thinking: if Player 2 likes utility, then Player 2 should prefer the C1 equilibrium. And Player 1 knows that Player 2 will have considered this. And vice versa. So, <C1,C1> is can be thought of as a “focal point,” something by which to coordinate behavior.

2 Things to Notice

Focal Points

Thomas Schelling’s (1960) experiment:Ask a group of respondents to imagine that they are one of two individuals unable to communicate, but needing to meet each other in New York City.

And Another Game…

(6,6) (1,10)

(10,1) (2,2)

Player 2

Player 1

C1

C2

C1 C2

What is the rational choice to make?

• C2 is the dominant strategy, and so C2 is the rational strategy.

Def: an outcome is Pareto Efficient (or “Pareto Optimal”) if there is no other outcome that gives at least one person a higher utility and give no one a lower utility. <C1,C1> is optimal.

C2 is rational, but not (Pareto) optimal. <C1,C1> is optimal.

This last game is the “Prisoner’s Dilemma.” Framed in terms of years in prison as follows:

(2,2) (10,1)

(1,10) (6,6)Player 1

Silent

Spill the beans

Silent Spill the beans

Player 2

According to standard assumptions of game theory, dominant strategies are rational, and therefore in the Prisoner’s Dilemma, the socially worst outcome is rational.

Garret Hardin. “The Tragedy of the Commons”. Science. 1968.

Solution: “mutual coercion, mutually agreed upon.”

Consider countries choosing either to abate their emissions of greenhouses gas emissions, or to “free-riding” on the effort of other countries…

Elinor Ostrom (1990): There is a meta-dilemma at play. Each country wishes that the others take up the costs of seeking the difficult-to-reach agreement on a coercion mechanism.

New dilemma: contribute to the common good of designing and implementing coercive mechanisms, or let other countries do it. This meta-dilemma generalizes for lots of social contract games like the prisoner’s dilemma.

As a fact, we do solve some prisoner’s dilemmas (speed limits). But how can we explain the evolution of this kind of cooperation within the standard model of rational choice theory.

Options:(1) We are simply acting (individually) irrational.

(2) We have “rationality” wrong.(3) We don’t need to change our model of a rational agent, because we selfishly prefer cooperation / fairness.

(Jon Gowdy is one of those pioneering a new understanding of what rationality entails.)

Cost-Benefit Analysis (CBA)Consider an imaginary air pollutant X. X is a by-product of

producing some important goods, but high levels of X is known to pose health concerns like asthma and cancer.

Extreme options: (1) Do nothing, no matter how much X is in the atmosphere? (2) Remove every last particle from the atmosphere?Rationality seems to demand that we make decisions at the margin: what is the quantity of X in the atmosphere above which we would be better off limiting X more?

(Invest in regulating X to the extent that it benefits society).

This is what economic cost-benefit analysis allows us to do.

Probabilities vs. Uncertainty

• According to rational choice theory, when probabilities of alternatives are known, we can multiply utilities by probabilities and decision-make in terms of expected utility rather than by utility.

• But sometimes probabilities are not known. Sometimes an outcome is only known to be in theory possible (full stop). What would rationality demand under uncertainty?

McKibben

• Chapter 2: assemblers! They could possibly destroy us all! But it will be individually rational for companies to try to develop these!

Can we justify regulation? Would such legislation be rational?

What’s the probability that assemblers will go horribly wrong?

Gardiner (2006) and Sunstein (2005)

“Anti-Catastrophe Principle”: Gardiner: Acting to eliminate a worst-case

scenario is the right strategy to adopt when (1) one cannot assign a probability (but only plausibility) to a potentially disastrous outcome and (2) the cost of acting is small.

Problems for Gardiner

(1) No new guidance here. (Of course eliminate a horrible scenario if you can do it for very little cost. This is trivial).

(2) World doesn’t really give us problems like this.

• Sunstein: when there is an uncertain danger of catastrophe, and when the costs of reducing those dangers are not huge and when incurring those costs does not divert resources from more pressing problems, then eliminate the danger.”

• 2 Problem:– Supposing the criteria are met, is it rational to

abandon developing assemblers, no matter what benefits are at stake?

ex: suppose we can know that assemblers will save millions of lives. The cost of NOT developing them is great now too. Great costs on both sides of the equation – what catastrophe should be avoided?ex: the principle would have us forgo relatively certain huge benefits because of the specter of possible huge costs. Seems irrational.