The Absent-Minded Driver

15 Pages·1997·0.17 MB·English

GAMESANDECONOMICBEHAVIOR20,102(cid:93)116(cid:14)1997.
ARTICLENO.GA970577
(cid:85)
The Absent-Minded Driver
RobertJ.Aumann,SergiuHart,andMottyPerry
(cid:67)(cid:101)(cid:110)(cid:116)(cid:101)(cid:114)(cid:102)(cid:111)(cid:114)(cid:82)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:97)(cid:108)(cid:105)(cid:116)(cid:121)(cid:97)(cid:110)(cid:100)(cid:73)(cid:110)(cid:116)(cid:101)(cid:114)(cid:97)(cid:99)(cid:116)(cid:105)(cid:168)(cid:101)(cid:68)(cid:101)(cid:99)(cid:105)(cid:115)(cid:105)(cid:111)(cid:110)(cid:84)(cid:104)(cid:101)(cid:111)(cid:114)(cid:121)(cid:44)(cid:84)(cid:104)(cid:101)(cid:72)(cid:101)(cid:98)(cid:114)(cid:101)(cid:119)(cid:85)(cid:110)(cid:105)(cid:168)(cid:101)(cid:114)(cid:115)(cid:105)(cid:116)(cid:121)(cid:111)(cid:102)
(cid:74)(cid:101)(cid:114)(cid:117)(cid:115)(cid:97)(cid:108)(cid:101)(cid:109)(cid:44)(cid:70)(cid:101)(cid:108)(cid:100)(cid:109)(cid:97)(cid:110)(cid:66)(cid:117)(cid:105)(cid:108)(cid:100)(cid:105)(cid:110)(cid:103)(cid:44)(cid:71)(cid:105)(cid:168)(cid:97)(cid:116)(cid:45)(cid:82)(cid:97)(cid:109)(cid:44)(cid:57)(cid:49)(cid:57)(cid:48)(cid:52)(cid:74)(cid:101)(cid:114)(cid:117)(cid:115)(cid:97)(cid:108)(cid:101)(cid:109)(cid:44)(cid:73)(cid:115)(cid:114)(cid:97)(cid:101)(cid:108)†
ReceivedFebruary6,1996
The example of the ‘‘absent-minded driver’’ was introduced by Piccione and
Rubinstein in the context of games and decision problems with imperfect recall.
Theyclaim that a ‘‘paradox’’or‘‘inconsistency’’ arises when the decision reached
at the ‘‘planning stage’’ is compared with that at the ‘‘action stage.’’ Though the
exampleis provocativeandworthhaving,their analysis is questionable. Acareful
analysis revealsthatwhiletheconsiderationsattheplanningandactionstagesdo
differ,thereisnoparadoxorinconsistency. (cid:74)(cid:111)(cid:117)(cid:114)(cid:110)(cid:97)(cid:108)(cid:111)(cid:102)(cid:69)(cid:99)(cid:111)(cid:110)(cid:111)(cid:109)(cid:105)(cid:99)(cid:76)(cid:105)(cid:116)(cid:101)(cid:114)(cid:97)(cid:116)(cid:117)(cid:114)(cid:101)Classi-
ficationNumbers:D81,C72. (cid:81)1997AcademicPress
1. INTRODUCTION
An absent-minded driver starts driving at START in Figure 1. At (cid:88) he
can either EXIT andgetto (cid:65)(cid:14)forapayoffof0.orCONTINUE to (cid:89).At (cid:89) he
can either EXIT and get to (cid:66) (cid:14)payoff 4., or CONTINUE to (cid:67) (cid:14)payoff 1.. The
essential assumption is that he cannot distinguish between intersections (cid:88)
and (cid:89), and cannot remember whether he has already gone through one of
them.
Piccione and Rubinstein (cid:14)1997; henceforth P&R., who introduced this
example, claim that a ‘‘paradox’’ or ‘‘inconsistency’’ arises when the
decision reached at the (cid:112)(cid:108)(cid:97)(cid:110)(cid:110)(cid:105)(cid:110)(cid:103) (cid:115)(cid:116)(cid:97)(cid:103)(cid:101)(cid:125)at START(cid:125)is compared with that
at the (cid:97)(cid:99)(cid:116)(cid:105)(cid:111)(cid:110) (cid:115)(cid:116)(cid:97)(cid:103)(cid:101)(cid:125)when the driver is at an intersection. Though the
example is provocativeand worth having, P&R’sanalysis seems flawed.A
careful analysis reveals that while the considerations at the planning and
action stages do differ, there is no paradox or inconsistency.
(cid:85)
Thisis an outgrowthofnotes andcorrespondenceoriginating in MayandJune of1994.
We thank Ehud Kalai, Roger Myerson, Phil Reny, Ariel Rubinstein, Dov Samet, Larry
Samuelson,andAsherWolinskyfordiscussionsonthesetopics,andtheAssociateEditorand
the referee for some insightful comments. Research partially supported by grants of the
U.S.(cid:93)IsraelBinationalScienceFoundation,theIsraelAcademyofSciencesandHumanities,
andtheGameTheoryProgramatSUNY(cid:93)StonyBrook.
†E-mail:[email protected].
102
0899-8256(cid:114)97$25.00
Copyright(cid:81)1997byAcademicPress
Allrightsofreproductioninanyformreserved.
THE ABSENT-MINDED DRIVER 103
FIG.1. Theabsent-mindeddriverproblem.
We start in Section 2 by laying down the fundamental observations that
underlie the driver’s decision problem, and then show in Section 3 how
P&R’s analysis violates these observations. In Section 4 we formally
define the concept of (cid:97)(cid:99)(cid:116)(cid:105)(cid:111)(cid:110)(cid:45)(cid:111)(cid:112)(cid:116)(cid:105)(cid:109)(cid:97)(cid:108)(cid:105)(cid:116)(cid:121) and use it to analyze P&R’s
example. Section 5 studies action-optimality in a more general setup, with
someinteresting and unexpected conclusions. We conclude with a detailed
discussion of various issues in Section 6.
2. FUNDAMENTALS
At the planning stage, the decision problem is straightforward. In the
example, the optimal (cid:14)randomized. decision is1‘‘CONTINUE with probability
2(cid:114)3 and EXIT with probability 1(cid:114)3.’’ We call this the (cid:112)(cid:108)(cid:97)(cid:110)(cid:110)(cid:105)(cid:110)(cid:103)(cid:45)(cid:111)(cid:112)(cid:116)(cid:105)(cid:109)(cid:97)(cid:108)
decision.
1The problem is to maximize (cid:14)1(cid:121)(cid:112).(cid:63)0(cid:113)(cid:112)(cid:14)1(cid:121)(cid:112).(cid:63)4(cid:113)(cid:112)2(cid:63)1 over (cid:112), where (cid:112) is the
probabilityofCONTINUE.
104 AUMANN, HART, AND PERRY
At the action stage, though, even (cid:102)(cid:111)(cid:114)(cid:109)(cid:117)(cid:108)(cid:97)(cid:116)(cid:105)(cid:110)(cid:103) the decision problem is
not straightforward. The following observations are essential for a correct
analysis of the decision at the action stage.2
(cid:118) First, the driver makes a decision at (cid:101)(cid:97)(cid:99)(cid:104) intersection through
which he passes. Moreover, when at one intersection, he can determine
the action (cid:111)(cid:110)(cid:108)(cid:121) (cid:116)(cid:104)(cid:101)(cid:114)(cid:101), and (cid:110)(cid:111)(cid:116) at the other intersection(cid:125)where he isn’t.
(cid:118) Second, since he is in completely indistinguishable situations at the
two intersections, whatever reasoning obtains at one must obtain also at
the other, and he is aware of this.
3. THE P&R ANALYSIS
Considerthe action stage.Thedriverfindshimself at an intersection; he
does not know which. Let (cid:97) be the probability that (cid:88) is the current
intersection, and let (cid:112) and (cid:113) be the probabilities of CONTINUE at the
current and at ‘‘other’’ intersections, respectively. Then the expected
payoff at the action stage is
(cid:72)(cid:14) (cid:112),(cid:113),(cid:97). (cid:91)(cid:97) (cid:14)1(cid:121)(cid:112). (cid:63)0(cid:113)(cid:112)(cid:14)1(cid:121)(cid:113). (cid:63)4(cid:113)(cid:112)(cid:113)(cid:63)1
(cid:113)(cid:14)1(cid:121)(cid:97). (cid:14)1(cid:121)(cid:112). (cid:63)4(cid:113)(cid:112)(cid:63)1 .
P&R maximize (cid:72)(cid:14)(cid:112), (cid:112),(cid:97).(cid:115)(cid:97)(cid:119)(cid:14)1(cid:121)(cid:112).(cid:63)0(cid:113)(cid:112)(cid:14)1(cid:121)(cid:112).(cid:63)4(cid:113)(cid:112)2(cid:63)1(cid:120)(cid:113)(cid:14)1
(cid:121)(cid:97).(cid:119)(cid:14)1(cid:121)(cid:112).(cid:63)4(cid:113)(cid:112)(cid:63)1(cid:120) over (cid:112), holding (cid:97) fixed. Thus they take (cid:112) and (cid:113)
as decision variables to be maximized simultaneously, subject to the
constraint (cid:113)(cid:115)(cid:112).Thismakessense onlyifthe drivercontrols the probabil-
ities at both intersections(cid:125)a violation of the first observation. But even if,
by some magical process, the driver (cid:99)(cid:111)(cid:117)(cid:108)(cid:100) control the probability (cid:113) at the
other intersection, surely (cid:97)depends on (cid:113), and cannot be held fixed in the
maximization!
4. ACTION-OPTIMALITY
How,then, (cid:115)(cid:104)(cid:111)(cid:117)(cid:108)(cid:100) the driverreason at the action stage?Let us spell out
in detail the implications of the two observations in Section 2:
(cid:14)i. The optimal decision is the same at both intersections; call
(cid:85)
it (cid:112) .
2Onecanimaginescenarios forwhichthese observationsdonothold.Butsuch scenarios
do not correspond to the plain meaning of the words used to describe the situation. More
important, with those other scenarios the analysis at the planning stage also changes, and
again there is no paradox.Piccione and Rubinstein have yet to adduce an explicit scenario
that (cid:100)(cid:111)(cid:101)(cid:115)displayaparadox.SeeSection6(cid:14)c.formoreonthisissue.
THE ABSENT-MINDED DRIVER 105
(cid:14)ii. Therefore, at each intersection, the driver believes that (cid:112)(cid:85) is
chosen at the other intersection.
(cid:14)iii. At each intersection, the driver optimizes his decision given his
(cid:85)
beliefs.Therefore,choosing (cid:112) atthecurrent intersection tobe (cid:112) mustbe
(cid:85)
optimal given the belief that (cid:112) is chosen at the other intersection.
Given a behavior (cid:113) at the other intersection, the probability that the
current intersection is (cid:88) is3 (cid:97)(cid:115)1(cid:114)(cid:14)1(cid:113)(cid:113).. Thus if we set (cid:104)(cid:14)(cid:112),(cid:113).(cid:91)
(cid:72)(cid:14)(cid:112),(cid:113),1(cid:114)(cid:14)1(cid:113)(cid:113).., we can restate the final implication (cid:14)iii. as follows:
(cid:112)(cid:85) is (cid:97)(cid:99)(cid:116)(cid:105)(cid:111)(cid:110)-(cid:111)(cid:112)(cid:116)(cid:105)(cid:109)(cid:97)(cid:108)ifthemaximumof (cid:104)(cid:14) (cid:112), (cid:112)(cid:85). over (cid:112)
isattainedat (cid:112)(cid:115)(cid:112)(cid:85).
Thus, (cid:112)(cid:85) is a fixed point4 of the (cid:14)set-valued. mapping (cid:113) (cid:170)
argmax (cid:104)(cid:14)(cid:112),(cid:113)..
(cid:112)
Applying this analysis to the example, we see that the (cid:14)randomized5.
planning-optimal decision(cid:125)CONTINUE with probability 2(cid:114)3(cid:125)is also ac-
tion-optimal. Indeed, if this is the behavior at the other intersection, then
the probability that the current intersection is (cid:88) is (cid:97)(cid:115)3(cid:114)5. Therefore
the expected payoff from choosing CONTINUE at the current intersection
with probability (cid:112) is (cid:104)(cid:14)(cid:112),2(cid:114)3.(cid:115)(cid:14)3(cid:114)5.(cid:63)(cid:119)(cid:14)1(cid:121)(cid:112).(cid:63)0(cid:113)(cid:112)(cid:63)(cid:14)1(cid:114)3.(cid:63)4(cid:113)(cid:112)(cid:63)
(cid:14)2(cid:114)3.(cid:63)1(cid:120)(cid:113)(cid:14)2(cid:114)5.(cid:63)(cid:119)(cid:14)1(cid:121)(cid:112).(cid:63)4(cid:113)(cid:112)(cid:63)1(cid:120), which equals 8(cid:114)5 (cid:14)for all (cid:112).. So
(cid:112)(cid:115)2(cid:114)3 maximizes it; thus (cid:112)(cid:85)(cid:115)2(cid:114)3 is action-optimal.
So there is no paradox: the planning-optimal choice of 2(cid:114)3 is also
action-optimal.
Moreover, (cid:112)(cid:85)(cid:115)2(cid:114)3 is the (cid:117)(cid:110)(cid:105)(cid:113)(cid:117)(cid:101) action-optimal decision. Indeed,
1
(cid:104)(cid:14) (cid:112),(cid:113). (cid:115) (cid:14)1(cid:121)(cid:112). (cid:63)0(cid:113)(cid:112)(cid:14)1(cid:121)(cid:113). (cid:63)4(cid:113)(cid:112)(cid:113)(cid:63)1
1(cid:113)(cid:113)
(cid:113)
(cid:113) (cid:14)1(cid:121)(cid:112). (cid:63)4(cid:113)(cid:112)(cid:63)1
1(cid:113)(cid:113)
(cid:14)4(cid:121)6(cid:113).(cid:112)(cid:113)4(cid:113)
(cid:115) .
1(cid:113)(cid:113)
3Thisprobabilityisthe‘‘consistentbelief’’ofP&R;butunlikeP&R,weconsiderittobe
(cid:116)(cid:104)(cid:101) one that is appropriate to this problem. In the Appendix we derive it from a formal
model.Informally,sincethedriveralwaysgoesthrough (cid:88),butonly (cid:113)ofthetimethrough(cid:89),
the ratio of the probabilities is 1 to (cid:113), so they must be 1(cid:114)(cid:14)1(cid:113)(cid:113). and (cid:113)(cid:114)(cid:14)1(cid:113)(cid:113).. These
probabilitiesmayalsobederivedfroma‘‘fairlottery’’approach(cid:14)seeFootnote3inAumann,
Hart,andPerry,1997..
4Formally, (cid:14)(cid:112)(cid:85),(cid:112)(cid:85). is a symmetric Nash equilibrium in the (cid:14)symmetric. game between
‘‘the driver at the current intersection’’ and ‘‘the driver at the other intersection’’ (cid:14)the
strategicformgamewithpayofffunctions (cid:104)..
5Forthepurecase,seeSection6(cid:14)d.below.
106 AUMANN, HART, AND PERRY
FIG.2. Multipleaction-optimaldecisions.
Given (cid:113),the maximizing (cid:112) therefore is: 0for (cid:113)(cid:41)2(cid:114)3;1for (cid:113)(cid:45)2(cid:114)3;and
anything for (cid:113)(cid:115)2(cid:114)3. Thus the only fixed point is (cid:112)(cid:85)(cid:115)2(cid:114)3.
The notion of action-optimality defined here is mathematically identical
to the ‘‘modifiedmultiselves approach’’described near the end (cid:14)Section 7.
of P&R. But unlike P&R, we consider this notion to be (cid:116)(cid:104)(cid:101) natural and
correct formulation of the driver’s decision problem at the action stage.
See Section 6 below for further discussion.
5. A MORE CHALLENGING EXAMPLE, WITH MULTIPLE
ACTION-OPTIMAL DECISIONS
We have seen that, in the specific example of P&R, randomized
planning optimality and action optimality coincide. This is not always so!
While, in general, any planning-optimal randomized decision is also
action-optimal,6 we will now show that there may be action-optimal
choices that are not planning-optimal. Surprisingly, there may even be
action-optimal choices that, at the intersections, look better than the
planning-optimal choice.
6Proposition3ofP&RandalsotheendoftheAppendixbelow.
THE ABSENT-MINDED DRIVER 107
FIG.3. Amorechallengingexample.
For a simple example with action-optimal decisions that are not plan-
ning-optimal,changethepayoffstobe1at (cid:65),0at (cid:66),and2at (cid:67)(cid:125)seeFig.
2. The unique planning-optimal choice is CONTINUE, i.e., (cid:112)(cid:115)1. There are,
however, three action-optimal decisions: (cid:112)(cid:85)(cid:115)(cid:112)(cid:115)1, (cid:112)(cid:85)(cid:115)0, and (cid:112)(cid:85)(cid:115)
1 2 3
1(cid:114)4 (cid:14)e.g.,to see that (cid:112)(cid:85)(cid:115)0 is action-optimal, note that if the decision at
2
the other intersection is EXIT, then it is indeed optimal to EXIT now too..
This leads us to the next point: When there are multiple action-optimal
choices, how do their payoffs compare? Of course, when computed at
START, the one that is planning-optimal yields the maximum payoff. But
how does it look at the current intersection? In the example above,
(cid:112)(cid:85)(cid:115)(cid:112)(cid:115)1 yields the highest payoff among all the action-optimal deci-
1
sions also when comparedat the intersections; i.e., (cid:104)(cid:14)(cid:112)(cid:85), (cid:112)(cid:85).(cid:41)(cid:104)(cid:14)(cid:112)(cid:85), (cid:112)(cid:85).
1 1 (cid:105) (cid:105)
for (cid:105)(cid:115)2,3. But this is not always so when there are more than two
intersections. Indeed, consider an example with three intersections, the
108 AUMANN, HART, AND PERRY
payoff being 7 at the first EXIT, 0 at the second EXIT, 22 at the third EXIT,
and 2 if always CONTINUE (cid:14)see Fig. 3.. Then:
(cid:14)i. The unique planning-optimal decision is (cid:112)(cid:115)0 (cid:14)with a payoff
of 7..
(cid:14)ii. There are three action-optimal decisions: (cid:112)(cid:85)(cid:115)(cid:112)(cid:115)0, (cid:112)(cid:85)(cid:115)
1 2
7(cid:114)30, and (cid:112)(cid:85)(cid:115)1(cid:114)2.
3
(cid:14)iii. The ex-ante expected payoffs for (cid:112)(cid:85), (cid:112)(cid:85), and (cid:112)(cid:85) are, respec-
1 2 3
tively, 7, 8519(cid:114)1350(cid:102)6.31, and 6.5.
(cid:14)iv. The ex-post expected payoffs (cid:104)(cid:14)(cid:112)(cid:85), (cid:112)(cid:85). for (cid:112)(cid:85), (cid:112)(cid:85),and (cid:112)(cid:85) are,
1 2 3
respectively, 7, 7378(cid:114)1159(cid:102)6.37, and 50(cid:114)7(cid:102)7.14; thus (cid:104)(cid:14)(cid:112)(cid:85), (cid:112)(cid:85). is
3 3
larger than (cid:104)(cid:14)(cid:112)(cid:85), (cid:112)(cid:85).(cid:39)(cid:104)(cid:14)(cid:112), (cid:112)..
1 1
The reader may ask, since the choice is among three possibilities
yielding 7, (cid:102)6.37, or (cid:102)7.14, why does the driver not choose the action
(cid:85)
with the highest yield, namely (cid:112) ? The answer, of course, is that at the
3
(cid:85) (cid:85) (cid:85)
action stage, the driver (cid:99)(cid:97)(cid:110)(cid:110)(cid:111)(cid:116) choose among (cid:112) , (cid:112) , and (cid:112) . His beliefs
1 2 3
there are not under his control; he cannot choose what to believe. Action
optimalityis a conditionfor consistency ofbeliefs and rational behavior.If
the player is to be consistent at the action stage, he must believe in one of
(cid:85) (cid:85) (cid:85)
the three possibilities (cid:112) , (cid:112) , (cid:112) ; but (cid:119)(cid:104)(cid:105)(cid:99)(cid:104) one is not up to him at that
1 2 3
stage.
We have already pointed to the formal similarity between action-opti-
(cid:85)
mality and game equilibria. Choosing (cid:98)(cid:101)(cid:116)(cid:119)(cid:101)(cid:101)(cid:110) the (cid:112) is much like choos-
(cid:105)
ing between game equilibria, which is something that the individual player
in a game cannot do(cid:125)it must be done by an outside force (cid:14)like a custom
or a norm., or by all the players somehow coordinating their actions.
In our case, there is only one player, who acts at different times.
Because of his absent-mindedness, he had better coordinate his actions;
this coordinationcan take placeonlybefore he starts out(cid:125)at the planning
(cid:85) (cid:85)
stage. At that point, he should choose (cid:112) . If indeed he chose (cid:112) , there is
1 1
(cid:85) (cid:85)
no problem. If by mistake he chose (cid:112) or (cid:112) , then that is what he should
2 3
do at the action stage. (cid:14)If he chose something else, or nothing at all, then
at the action stage he will have some hard thinking to do..
(cid:85)
Once having coordinated on (cid:112) , there is no incentive for the driver to
1
(cid:85)
choose (cid:112) at the action stage. Nevertheless, one may ask whether at that
3
(cid:85) (cid:85)
stage he will be sorry that he didnot coordinate on (cid:112) rather than on (cid:112) .
3 1
After all, if he had,his expectation nowwouldbe (cid:102)7.14,which is greater
than the 7 he now expects! If the answer is ‘‘yes,’’ we have a conceptual
puzzle; why should the driver get himself into a situation where at START
he is (cid:115)(cid:117)(cid:114)(cid:101) that he will be sorry at every intersection he reaches? Why not
(cid:85)
avoid the sorrow by coordinating on (cid:112) in the first place?
3
THE ABSENT-MINDED DRIVER 109
FIG.4. Anautomaticcar.
(cid:85)
But the answer is ‘‘no’’; he should not be sorry. Having chosen (cid:112) , he
1
knows he must be at (cid:88) when finding himself at an intersection. Being at
(cid:88) is like being at START(cid:125)i.e., at the planning stage(cid:125)and then the best
(cid:85) (cid:85)
choice is (cid:112) and (cid:111)(cid:110)(cid:108)(cid:121) (cid:112) . So when reaching an intersection after having
1 1
(cid:85) (cid:85) (cid:85)
chosen (cid:112) ,the driveris (cid:110)(cid:111)(cid:116) sorrythat he indeedchose (cid:112) rather than (cid:112) .
1 1 3
Why, then, is the driver’s expectation at an intersection nevertheless
(cid:85) (cid:85)
larger for (cid:112) than for (cid:112) ? The reason is that at an intersection, his belief
3 1
(cid:85) (cid:85)
as to where he is if he chose (cid:112) differs from his belief when he chose (cid:112) .
3 1
(cid:85) (cid:85)
Having chosen (cid:112) , he knows he must be at (cid:88). If he had chosen (cid:112) , he
1 3
would have attributed probabilities 4(cid:114)7, 2(cid:114)7, and 1(cid:114)7 to being at (cid:88), (cid:89),
and (cid:90). He ‘‘prefers’’ the latter distribution, because it gives him a chance
of already having passed the ‘‘dangerous’’ intersection (cid:89) and a better shot
at the high payoff of 22. But as we said above, one cannot choose one’s
beliefs, and it makes little sense to discuss ‘‘preferences’’ between them.
Specifically,since he doesknowthat he is at (cid:88),it wouldbe silly forhimto
say, ‘‘I wish I had chosen the other plan, because then in my ignorance I
would have been deluded into expecting a higher payoff than now.’’
Toclarify this point,considerthe exampleinFig.4.Thecar isautomatic
and EXITS with probability 1(cid:114)2 at each intersection. The decision maker is
110 AUMANN, HART, AND PERRY
FIG.5. Clear-headedorabsent-minded?
a passenger, who sleeps during most of the trip. At START, he is given the
option to be woken either at both intersections, or only at (cid:88). In the first
option he is absent-minded: when waking up, he does not know at which
intersection he is. We call the second option ‘‘clear-headedness.’’
As in the previous discussion, the question at (cid:88) is not operative(cid:125)what
to do(cid:125)but only whether it makes sense to be ‘‘sorry.’’ If he chose
clear-headedness, his expectation upon reaching (cid:88) is 1(cid:114)4. If he had
chosen absent-mindedness, then when reaching (cid:88) he would have at-
tributed probability 2(cid:114)3 to being at (cid:88) and 1(cid:114)3 to being at (cid:89). Therefore
his expected payoff at that point would have been (cid:14)2(cid:114)3.(cid:63)(cid:14)1(cid:114)4.(cid:113)(cid:14)1(cid:114)3.(cid:63)
(cid:14)1(cid:114)2.(cid:115)1(cid:114)3, which is larger than 1(cid:114)4. Is he therefore sorry that he chose
to be clear-headed? Clearly this would be absurd, as the payoffs do not
depend on his choice.
To make this even more striking, assume that when he is not absent-
minded, the probability of CONTINUE is increased to 4(cid:114)7 (cid:14)Fig. 5.. Then
ex-ante the clear-headed optionis actually preferred to the absent-minded
option(cid:125)it yields 16(cid:114)49 rather than 1(cid:114)4. But, upon reaching (cid:88), the
clear-headed option still yields 16(cid:114)49, whereas the absent-minded option
THE ABSENT-MINDED DRIVER 111
yields 1(cid:114)3 (cid:14)as above., which is bigger then 16(cid:114)49. Surely, it would be
absurd for the decision maker to wish he were absent-minded(cid:125)it would
be sticking his head in the sand!
Another issue that these examples raise is this7: Assume the driver in
(cid:85)
Fig. 3 has chosen (cid:112) at START. His expected payoff there is then 6.5. But,
3
he is (cid:99)(cid:101)(cid:114)(cid:116)(cid:97)(cid:105)(cid:110) to gothrough (cid:88), where his expected payoff becomes (cid:102)7.14.
For a treatment of this issue(cid:125)which is entirely different from that of the
current paper(cid:125)please see Aumann, Hart, and Perry (cid:14)1997..
6. DISCUSSION
This section elaborates on a number of different matters.
(cid:14)a. (cid:68)(cid:101)(cid:99)(cid:105)(cid:115)(cid:105)(cid:111)(cid:110) (cid:80)(cid:111)(cid:105)(cid:110)(cid:116)(cid:115). Part of the problem in P&R’s analysis is in their
interpretation ofinformation sets. Recall that the extensive formdescribes
the waya gameis (cid:112)(cid:108)(cid:97)(cid:121)(cid:101)(cid:100).Theplayproceedsfromone nodetothe next, as
each player is called upon to make a choice whenever a decision node of
his is reached. Of course, when asked to make a choice, he possesses
certain information. This is accurately described by information sets: two
decision nodes where a player’s information is the same belong to the
same information set. But decisions are madeat (cid:110)(cid:111)(cid:100)(cid:101)(cid:115), (cid:110)(cid:111)(cid:116) at information
sets (cid:14)cf. the first observation of Section 2..
Ingameswithperfectrecall,agiveninformationsetcanbereachedonly
once(cid:125)at a single node(cid:125)in any one play of the game. Therefore, there is
noharmin identifying decisionpointswithinformationsets in such games,
thougheventhere the decisionpointis basically anode.But in gameswith
absent-mindedness, when an information set may be visited more than
once,itissimplyincorrect toidentifydecisionpointswithinformationsets.
(cid:14)b. (cid:67)(cid:111)(cid:110)(cid:116)(cid:114)(cid:111)(cid:108). At each intersection, the driver ‘‘expects’’ that he will do
the same at the other intersection. He (cid:101)(cid:120)(cid:112)(cid:101)(cid:99)(cid:116)(cid:115) it, and maximizes given that
expectation, and the maximizing behavior (cid:116)(cid:117)(cid:114)(cid:110)(cid:115) (cid:111)(cid:117)(cid:116) the same as the
expectation; that is precisely action-optimality. But (cid:101)(cid:120)(cid:112)(cid:101)(cid:99)(cid:116)(cid:105)(cid:110)(cid:103) is not (cid:100)(cid:101)(cid:116)(cid:101)(cid:114)-
(cid:109)(cid:105)(cid:110)(cid:105)(cid:110)(cid:103). He cannot, in fact, determine it(cid:125)he cannot (cid:99)(cid:111)(cid:110)(cid:116)(cid:114)(cid:111)(cid:108) what happens
at the other intersection.
IntheirSection7,P&Rdiscusswhattheycallthe‘‘modifiedmulti-selves
approach,’’ which is the same as our notion of action-optimality. They
write that this approach‘‘(cid:97)(cid:115)(cid:115)(cid:117)(cid:109)(cid:101)(cid:115) that a decision maker,uponreaching an
information set, takes his actions tobe immutable at future occurrences of
that information set....Atthe other extreme one findsthe oppositeaxiom
7We are grateful to the associate editor in charge of the current paper for raising this
question.