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  • If b is the event that a voter goes for Bob, the event that Pandoras vote is piv-otal after receiving the message a can be represented as abb. To make her decision,Pandora needs to use Bayess rule to nd the larger of the conditional proba-


    prob (A j abb) c prob (abb jA) prob (A) c 23

    23p 1


    2 12;

    prob (B j abb) c prob (abb jB) prob (B) c 13

    13p 2


    2 12:

    We consider two cases. In the rst, Pandora knows that the other free voters

    wont notice that their vote can matter only when they are pivotal. They therefore

    simply vote for whichever candidate is favored by their own message. Thus p 0,and so prob (A j abb)< prob (B j abb). It follows that Pandora should vote for Bob allthe timeeven when her own message favors Alice! If this outcome seems para-

    doxical, reect that Pandora will be pivotal in favor of Alice only when the two other

    free voters have received messages favoring Bob. The messages will then favor Bob

    by two to one.

    The second case arises when it is common knowledge that all the free voters are

    game theorists. To nd a symmetric equilibrium in mixed strategies we simply set

    prob (A j abb) prob (B j abb), which happens when p 0.32 (Exercise 13.10.8.).Pandora will then vote for Bob slightly less than a third of the time when her own

    message favors Alice.

    Critics of game theory dont care for this kind of answer. Strategic voting is bad

    enough, but randomizing your vote is surely the pits! However, Immanuel Kant is

    on our side for once. If everybody except Alices parents votes like a game theorist,

    the better candidate is elected with a probability of about 0.65. If everybody except

    Alices parents votes for the candidate favored by their own message, not only is the

    outcome unstable, but the better candidate is elected with a probability of about 0.63

    (Exercise 13.10.9).

    13.3 Bayesian Rationality

    If Bayesian decision theory consisted of just updating probabilities using Bayess

    rule, there wouldnt be much to it. But it also applies when we arent told what

    probabilities to attach to future events. This section explains how Von Neumann and

    Morgensterns theory can be extended to cover this case.

    13.3.1 Risk and Uncertainty

    Economists say they are dealing with risk when the choices made by Chance come

    with objectively determined probabilities. Spinning a roulette wheel is the arche-


    ! 13.4

    2Note that prob (abb jA) prob (a jA){prob (b jA)}2. Also, prob (a jA) 23

    and prob (b jA) prob (b j a jA) prob (a jA) prob (b j b jA) prob (b jA) p 2

    3 1 1

    3. We dont need to nd c. If we did,

    we could use the fact that c1 prob (abb) prob (abb jA) prob (abb jB) or the equation prob(A j abb) prob (B j abb) 1.

    388 Chapter 13. Keeping Up to Date

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  • typal example. On a standard wheel, the ball is equally likely to stop in any of one of

    thirty-seven slots labeled 0, 1, . . . , 36. The fact that each slot is equally likely can beveried by observing the frequency with which each number wins in a very large

    number of spins. These frequencies are the data on which we base our estimates of

    the objective probability of each number. For example, if the number seven came up

    fty times in one hundred spins, everybody would become suspicious of the casinos

    claim that its probability is only 137.

    Economists speak of uncertainty when they dont want to claim that there is

    adequate objective data to tie down the probabilities with which Chance moves.

    Sometimes they say that such situations are ambiguous because different people

    might argue in favor of different probabilities. Betting on horses is the archetypal


    One cant observe the frequency with which Punters Folly will win next years

    Kentucky Derby because the race will be run only once. Nor do the odds quoted

    by bookies tell you the probabilities with which different horses will win. Even if

    the bookies knew the probabilities, they would skew the odds in their favor. Nev-

    ertheless, not only do people bet on horses, but they also go on blind dates. They

    change their jobs. They get married. They invest money in untried technologies.

    They try to prove theorems. What can we say about rational choice in such uncertain


    Economists apply a souped-up version of the theory of revealed preference de-

    scribed in Section 4.2. Just as Pandoras purchases in a supermarket can be regarded

    as revealing her preferences, so also can her bets at the racetrack be regarded as

    revealing both her preferences and her beliefs.

    13.3.2 Revealing Preferences and Beliefs

    A decision problem is a function f :AB!C that assigns a consequence c f (a, b)in C to each pair (a, b) in AB (Section 12.1.1). If Pandora chooses action a whenthe state of the world happens to be b, the outcome is c f (a, b). Pandora knows thatB is the set of states that are currently possible. Her beliefs tell her which possible

    states are more or less likely.

    Let a be the action in which Pandora bets on Punters Folly in the Kentucky

    Derby. Let E be the event that Punters Folly wins and E the event that it doesnt.The consequenceL f (a;E) represents what will happen to Pandora if she loses.The consequenceW f (a;E) represents what will happen if she wins.

    All this can be summarized by representing the action a as a table:

    EEa 13:1

    Such betting examples show why an act a can be identied with a functionG :B!Cdened by cG (b) f (a, b). When thinking of an act in this way, we call it agamble.

    Von Neumann and Morgensterns theory doesnt apply to horse racing because

    the necessary objective probabilities for the states of the world are unavailable,

    13.3 Bayesian Rationality 389

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  • but the theory can be extended from the case of risk to that of uncertainty by replacing

    the top line of Figure 4.6 by:


    . . .

    . . .

    E2 E3 En

    w1 w2 w3 wn

    p1 . . .

    . . .

    p2 p3 pn

    w1 w2 w3 wn 13:2

    The new line simply says that Pandora treats any gamble G as though it were a

    lottery L in which the probabilities pi prob (Ei) are Pandoras subjective proba-bilities for the events Ei.

    If Pandoras subjective probabilities pi prob (Ei) dont vary with the gamble G,we can then follow the method of Section 4.5.2 and nd her a Von Neumann and

    Morgenstern utility function u : O! R. Her behavior can then be described bysaying that she acts as though maximizing her expected utility

    Eu(G) p1u(o1) p2u(o2) pnu(on)

    relative to a subjective probability measure that determines pi prob (Ei).Bayesian rationality consists in separating your beliefs from your preferences in

    this particular way. Game theory assumes that all players are Bayesian rational. All

    that we need to know about the players is therefore summarized by their Von

    Neumann and Morgenstern utilities for each outcome of the game and their sub-

    jective probabilities for each chance move in the game.

    13.3.3 Dutch Books

    Why would Pandora behave as though a gamble G were equivalent to a lottery L?

    How do we nd her subjective probability measure? Why should this probability

    measure be the same for all gambles G?

    To appeal to a theory of revealed preference, we need Pandoras behavior to be

    both stable and consistent. Consistency was defended with a money-pump argument

    in Section 4.2.1. When bets are part of the scenario, we speak of Dutch books rather

    than money pumps.

    For an economist, making a Dutch book is the equivalent of an alchemist nding

    the fabled philosophers stone that transforms base metal into gold. But you dont

    need a crew of nuclear physicists and all their expensive equipment to make the

    economists stone. All you need are two stubborn people who differ about the

    probability of some event.

    Suppose that Adam is quite sure that the probability of Punters Folly winning the

    Kentucky Derby is 34. Eve is quite sure that the probability is only 1

    4. Adam will then

    accept small enough bets at any odds better than 1 : 3 against Punters Folly winning.

    Eve will accept small enough bets at any odds better than 1 : 3 against Punters Folly

    losing.3 A bookie can now make a Dutch book by betting one cent with Adam at

    odds of 1 : 2 and one cent with Eve at odds of 1 : 2. Whatever happens, the bookie

    loses one cent to one player but gets two cents from the other.


    ! 13.4

    3Assuming they have smooth Von Neumann and Morgenstern utility functions.

    390 Chapter 13. Keeping Up to Date

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  • This is the secret of how bookies make money. Far from being the wild gamblers

    they like their customers to think, they bet only on sure things.

    Avoiding Dutch Books. To justify introducing subjective probabilities in Section

    13.3.2, we need to assume that Pandoras choices reveal full and rational preferences

    over a large enough set of gambles (Section 4.2.2).

    Having full preferences will be taken to include the requirement that Pandora

    never refuses a betprovided that she gets to choose which side of the bet to back,

    which means that she chooses whether to be the bookie offering the bet or the

    gambler to whom the bet is offered. Being rational will simply mean th