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Unformatted text preview: More on Sequential and Simultaneous
Move Games
• So far we have studied two types of games: 1)
sequential move (extensive form) games where
sequential move (extensive form) games where
players take turns choosing actions and 2)
strategic form (normal form) games where players
simultaneously choose their actions.
• Of course it is possible to combine both game
forms as for example happens in the game of
forms as, for example, happens in the game of
football.
• The transformation between game forms may
change the set of equilibria, as we shall see.
• We will also learn the concept of subgame
perfection Combining Sequential and Simultaneous
Combining Sequential and Simultaneous
Moves.
• Consider the following 2 player game, where Player 1
the following player game, where Player
moves first:
Player 1
Stay Out Player 2 Enter A
3,3 Player 1 A
B B 2,2 3,0 0,3 4,4 • If player 1 chooses to stay out, both he and player 2 earn
a payoff of 3 each, but if player 1 chooses to enter, he
plays a simultaneous move game with player 2. Forward Induction
• The simultaneous move game has 3 equilibria: (A,A), (B,B) and a
mixed strategy equilibrium where both players play A with probability
1/3 and earn expected payoff 8/3. Player 1
Stay Out Player 2 Enter A
3,3
• • Player 1 A
B B 2,2 3,0 0,3 4,4 If player 2 sees that player 1 has chosen to Enter, player 2 can use
forward induction reasoning: since player 1 chose to forego a payoff of
3, it is likely that he will choose B, so I should also choose B.
lik
The likely equilibrium of the game is therefore: Enter, (B,B). The Incumbent
The IncumbentRival Game in Extensive and
Game in Extensive and
Strategic Form How many equilibria are
there in the extensive form of
this game? How many equilibria are
there in the strategic form of this
game? The Number of Equilibria Appears to be Different! There appears to be just 1
pp
equilibrium using rollback on the
extensive form game. There appears to be 2 equilibria
using cellbycell inspection of
the strategic form game Subgame Perfection
Subgame Perfection
• In the strategic form game, there is the additional
equilibrium Stay Out Fight that is not an equilibrium
equilibrium, Stay Out, Fight that is not an equilibrium
using rollback in the extensive form game.
• Equilibria found by applying rollback to the extensive
form game are referred to as subgame perfect equilibria:
every player makes a perfect best response at every
subgame of the tree.
– Enter, Accommodate is a subgame perfect equilibrium.
– Stay Out, Fight is not a subgame perfect equilibrium. • A subgame is the game that begins at any node of the
th
th
th
decision tree.
3 subgames (circled)
are all the games
beginning at all tree
nodes including the
root node (game itself) Imperfect Strategies are Incredible
• Strategies and equilibria that fail the test of subgame perfection
are called imperfect.
• The imperfection of a strategy that is part of an imperfect
Th
th
equilibrium is that at some point in the game it has an
unavoidable credibility problem.
• Consider for example, the equilibrium where the incumbent
promises to fight, so the rival chooses stay out.
• The incumbent’s promise is incredible; the rival knows that if he
enters, the incumbent is sure to accommodate, since if the
incumbent adheres to his promise to fight, both earn zero, while if
the incumbent accommodates, both earn a payoff of 2.
• Thus, Stay Out, Fight is a Nash equilibrium, but it is not a
subgame perfect Nash equilibrium.
• Lesson: Every subgame perfect equilibrium is a Nash equlibrium
eq
is Nash eq
but not every Nash equlibrium is a subgame perfect equilibrium! Another Example:
Mutually Assured Destruction (MAD)
What is
the rollback
(subgame perfect)
equilibrium
to this game? A subgame Subgames must
contain all nodes in an
information set
set Another
subgame The Strategic Form Version of the Game Admits 3
Nash Equilibria
• Which Equilibrium is Subgame Perfect? • Only the equilibrium where the strategies Escalate, Back Down are
played by both the U.S. and Russia is subgame perfect. – Why? From Simultaneous to Sequential Moves
Si
• Conversion from simultaneous to sequential moves
from simultaneous to sequential moves
involves determining who moves first, which is not
an issue in the simultaneous move game.
• In some games, where both players have dominant
strategies, it does not matter who moves first.
– For example the prisoner’s dilemma game.
th
dil • When neither player has a dominant strategy, the
subgame
subgame perfect equilibrium will depend on the
equilibrium will depend on the
order in which players move.
– For example, the Senate Race Game, the Pittsburgh LeftTurn Game. The Equilibrium in Prisoner’s Dilemma is the
Same Regardless of Who Moves First
Wh
Fi
This simultaneous
move game is
equivalent to
either of the
either of the 2
sequential move
games below it. The Senate Race Game has a Different Subgame
Perfect Equilibrium Depending on Who moves first.
Wh
fi
In the simultaneous
move game, there
is only one Nash
equilibrium. subgame perfect eq, subgame perfect eq, Similarly, in the Pittsburgh LeftTurn Game
Driver 1
Driver 2 Driver 2 Driver 1
Proceed
Driver 2
Proceed
1490, 1490 Driver 1 Driver 2 Yield Proceed
5, 5 Proceed Yield 5, 5 Yield
10, 10 Proceed
1490, 1490
1490 Yield
Driver 1 Yield Proceed
5, 5 5, 5 Yield
10, 10
10 These subgame perfect equilibria look the same, but if Driver
1 moves first he gets a payoff of 5, while if Driver 2 moves
moves first he gets payoff of while if Driver moves
first Driver 1 gets a payoff of –5, and vice versa for Driver 2. Going from a Simultaneous Move to a
Sequential Move Game may eliminate
the play of a mixed strategy equilibrium
• This is true in games with a unique mixed
strategy Nash equilibrium.
– Example: The Tennis Game Serena
Williams DL
CC Venus Williams
DL
CC
50, 50
80, 20
90, 10
20, 80 The Pure Strategy Equilibrium is Different
The Pure Strategy Equilibrium is Different,
Depending on Who Moves First. There is no possibility of mixing in a sequential
move game without any information sets. Empirical Plausibility
of Subgame
of Subgame Perfection?
• Consider the “Centipede Game”:
The original
game has 100
decision nodes
hence, the name.
For our purposes,
our purposes,
four nodes will
suffice. Unique Subgame Perfect
Equilibrium
• • • Player 1
chooses
Take at the
First
Opportunity.
Empirically
it takes time
for players to
fi
figure this
thi
equilibrium
prediction
out.
Players are
boundedly
rational Econometrica, Vol. 60, No. 4 (July, 1992), 803836 AN EXPERIMENTALSTUDY OF THE CENTIPEDEGAME
BY RICHARD D. MCKELVEY AND THOMASR. PALFREY 1 i
p
We reporton an experiment n whichindividuals laya versionof the centipedegame.
In this game, two players alternatelyget a chance to take the larger portion of a
continuallyescalatingpile of money. As soon as one person takes, the game ends with
that playergettingthe largerportionof the pile, and the other playergettingthe smaller
g
portion.If one views the experimentas a completeinformation ame, all standardgame
theoretic equilibrium oncepts predictthe first movershould take the large pile on the
c
firstround.The experimental esultsshow that this does not occur.
r
An alternativeexplanationfor the data can be given if we reconsiderthe game as a
game of incomplete informationin which there is some uncertaintyover the payoff
functions of the players. In particular,if the subjects believe there is some small
o
likelihoodthat the opponent is an altruist,then in the equilibrium f this incomplete
information ame, playersadopt mixedstrategiesin the early roundsof the experiment,
g
h
with the probability f takingincreasing s the pile gets larger.We investigate owwell a
o
a
versionof this model explainsthe data observedin the centipedeexperiments.
a
KEYWORDS:Game theory,experiments, ationality, ltruism.
r
A
1. OVERVIEW FTHEEXPERIMENT ND THERESULTS
O
T
o
THIS PAPERREPORTS HE RESULTS f several experimentalgames for which the
predictions of Nash equilibrium are widely acknowledgedto be intuitively
W
unsatisfactory. e explainthe deviationsfromthe standardpredictionsusing an
approachthat combinesrecent developmentsin game theorywith a parametric
specificationof the errors individualsmight make. We construct a structural
econometricmodel and estimatethe extent to which the behavioris explainable
by gametheoreticconsiderations.
In the games we investigate, the use of backwardinduction and/or the
eliminationof dominatedstrategiesleads to a uniqueNash prediction,but there
are clear benefits to the playersif, for some reason, some playersfail to behave
in this fashion.Thus, we have intentionallychosen an environmentin which we
t
expect Nash equilibrium o performat its worst. The best known exampleof a
game in this class is the finitely repeated prisoners'dilemma.We focus on an
even simpler and, we believe more compelling, example of such a game, the
closely related alternatingmovegame that has come to be known as the
"centipedegame"(see Binmore(1987)).
The centipede game is a finite move extensiveform two persongame in which
each playeralternatelygets a turn to either terminatethe game with a favorable
payoff to itself, or continue the game, resultingin social gains for the pair. As
'Supportfor this researchwas providedin partby NSF Grants#IST8513679and #SES878650
to the CaliforniaInstituteof Technology.We thank MahmoudElGamalfor valuablediscussions
a
t
e
concerning he econometric stimation, ndwe thankRichardBoylan,MarkFey, ArthurLupia,and
David Schmidtfor able researchassistance.We thankthe JPLCaltechoint computingprojectfor
j
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grantingus time on the CRAY XMP at the Jet PropulsionLaboratory. e also are gratefulfor
f
commentsand suggestionsfrom many seminar participants, rom an editor, and from two very
thoroughreferees.
803 806 R. D. MCKELVEY AND T. R. PALFREY niques. This enables us to estimate the numberof subjectsthat actuallybehave
in such a fashion, and to address the question as to whether the beliefs of
subjectsare on averagecorrect.
c
Our experiment an also be comparedto the literatureon repeatedprisoner's
dilemmas. This literature (see e.g., Selten and Stoecker (1986) for a review)
finds that experienced subjects exhibit a pattern of "tacit cooperation"until
shortlybefore the end of the game, when they start to adopt noncooperative
behavior.Such behaviorwould be predictedby incompleteinformationmodels
like that of Kreps et al. (1982). However, Selten and Stoecker also find that
inexperiencedsubjectsdo not immediatelyadopt this patternof play, but that it
takes them some time to "learn to cooperate."Selten and Stoecker develop a
learningtheory model that is not based on optimizingbehaviorto account for
such a learningphase. One could alternatively evelop a model similarto the
d
one used here, where in additionto incompleteinformation boutthe payoffsof
a
others, all subjects have some chance of making errors,which decreases over
time. If some other subjects might be making errors, then it could be in the
interest of all subjectsto take some time to learn to cooperate, since they can
masqueradeas slow learners. Thus, a natural analog of the model used here
might offer an alternativeexplanationfor the data in Selten and Stoecker.
2. EXPERIMENTAL DESIGN Our budget is too constrainedto use the payoffsproposedby Aumann.So we
run a rather more modest version of the centipede game. In our laboratory
games, we start with a total pot of $.50 divided into a large pile of $.40 and a
small pile of $.10. Each time a playerchooses to pass, both piles are multiplied
by two. We considerboth a two round(four move) and a three round(six move)
version of the game. This leads to the extensive forms illustratedin Figures 1
and 2. In addition,we consider a version of the four move game in which all
payoffs are quadrupled.This "high payoff" condition therefore produced a
payoffstructureequivalentto the last four moves of the six move game.
1 1T fT
0.40
0.10 1 2 0.20
0.80 FIGURE 1.The 2 I
.
fT
0.40
0.10 P P 14 2 6.40 fT fT 1.60
0.40 P 1.60 0.80
3.20 four movecentipedegame.
1 2
P P0 2 25.60
P P fT fT fT fT fT 0.20
0.80 1.60
0.40 0.80
3.20 6.40
1.60 3.20
I2.80 2.The six movecentipedegame.
FIGURE 6.40
26 808 R. D. MCKELVEYAND T. R. PALFREY
TABLE IIA
PROPORTION OF OBSERVATIONS AT EACH TERMINAL NODE f f7 .20
.35
.23 .01
.11
.12 .01
.02
.01 .253 .078 .014 N fi f2 f3 f4 f5 1
2
3 (PCC)
(PCC)
(CIT) 100
81
100 .06
.10
.06 .26
.38
.43 .44
.40
.28 .20
.11
.14 .04
.01
.09 Total 13 281 .071 .356 .370 .153 .049 High Payoff 4 (HighCIT) 100 .150 .370 .320 .110 .050 Six
Move 5
6
7 (CIT)
(PCC)
(PCC) 100
81
100 .02
.00
.00 .09
.02
.07 .39
.04
.14 .28
.46
.43 Total 57 281 .007 .064 .199 .384 Session Four
Move involved the high payoff four move game, and sessions 57 involved the six
move version of the game. This gives us a total of 58 subjectsand 281 plays of
the four move game, and 58 subjectswith 281 plays of the six move game, and
20 subjectswith 100 plays of the high payoff game. Subjectswere paid in cash
the cumulativeamount that they earned in the session plus a fixed amountfor
showingup ($3.00 for CIT students and $5.00 for PCC students).7
3. DESCRIPTIVE SUMMARY OF DATA The complete data from the experimentis given in AppendixC. In Table II,
t
we present some simple descriptivestatistics summarizing he behaviorof the
subjects in our experiment. Table IIA gives the frequencies of each of the
terminal outcomes. Thus fi is the proportion of games ending at the ith
terminalnode. Table IIB gives the implied probabilities, i of takingat the ith
p
decisionnode of the game. In other words,pi is the proportionof games among
those that reached decision node i, in which the subjectwho moves at node i
chose TAKE. Thus, in a game with n decision nodes, pi =fi/El f1f.
E
i
All standardgame theoretic solutions(Nash equilibrium, terated elimination
e
of dominatedstrategies,maximin,rationalizability, tc.) would predict fi = 1 if
i = 1, fI = 0 otherwise. The requirement of rationality that subjects not adopt dominatedstrategieswould predictthat fn+i = 0 and Pn= 1. As is evidentfrom
Table II, we can reject out of hand either of these hypothesesof rationality.In
only 7% of the four move games, 1% of the six move games, and 15% of the
high payoffgames does the firstmoverchoose TAKE on the firstround.So the
subjectsclearlydo not iterativelyeliminatedominatedstrategies.Further,when
a game reaches the last move, Table IIB shows that the player with the last
move adopts the dominatedstrategyof choosingPASS roughly25% of the time
7The stakes in these games were large by usual standards. Students earned from a low of $7.00 to
a high of $75.00, in sessions that averaged less than 1 houraverage earnings were $20.50 ($13.40 in
the four move, $30.77 in the six move, and $41.50 in the high payoff four move version). Strategic Moves
Strategic Moves
•
• So far we have supposed that the rules of the game were fixed, e.g., who
moves first, the timing of decisions, payoffs, etc.
In real strategic situations, players will have incentives to attempt to
manipulate the rules of the game/action choices/payoffs available for
their own benefit.
– E.g. who moves first, what choices remain, etc. • •
• A strategic move is an action taken outside the rules of the game
effectively transforming the original game, into a twostage game—such
moves are sometimes called “gamechangers.”
Some kind of strategic move is made in stage one and some version of
the original game, possibly with altered payoffs, is then played in stage 2.
For strategic moves to work, they must (1) be observable to the other
players and (2) irreversible (to the extent this is credible) so that they
alter other player’s expectations and make the outcome more favorable to
the player making the strategic move. Kinds of Strategic Moves
• • • • Commitments:
– Irreversibly limit your choice of action thereby forcing the other player to
choose his/her best response to your preferred action.
hi
– E.g., in the Pittsburgh leftturn game, I hit the gas pedal hard and jump the
light early to make my left turn, so your best response is to yield.
Threats: If you do not choose an action I prefer, I will respond in a manner
If you do not choose an action prefer will respond in manner
that will be bad for you (in the second stage).
– E.g., if you vote for ObamaCare, I will raise money for your opponent this
November.
Promises: If you choose an action I prefer, I will respond in a manner that will
be good for you.
– E.g., if you vote for the Senate health care bill, I will send a check for your
reelection campaign.
Note that both threats and promises are costly if they have to be carried out.
However, if a threat works to alter the target player’s behavior, there is no cost
to the player of issuing the threat, while if a promise works, there is a cost.
th
th th
if
th Wh St
Why Strategic Moves?
• Freedom of choice / a larger action space can
of choice larger action space can
be a bad thing.
– For example, allowing students to make up a
example allowing students to make up
midterm exam or perform extra credit is costly –
a new exam, extra assignments have to be
written/graded.
– By announcing an adhering to a no makeup, no
extra credit, no exceptions policy and announcing
it on day 1 of class, writing it into the syllabus,
etc. your professor is made better off. Credibility of Strategic Moves
Credibility of Strategic Moves
• The problem with strategic moves, especially threats
and promises is that they may not be credible
and promises is that they may not be credible.
– Expost, you may not want to carry out a costly threat or
follow through with a promised reward. • Strategic moves that are not credible will be ignored.
way to make strategic moves credible is to
• The way to make strategic moves credible is to
either take options off the table completely by
making a truly irreversible move, or make it costly
for the strategic mover to deviate from his strategic
move, i.e., change the payoffs of the game so that
following through with the strategic move is a best
th
th
response in the second stage. Illustration of Credibility Issue:
Nuisance Lawsuits
• Consider a game between a Plaintiff and a Defendant.
• Plaintiff moves first, deciding whether to file a lawsuit against
Defendant at cost to Plaintiff of k>0 If lawsuit is filed the
Defendant at cost to Plaintiff of k>0. If a lawsuit is filed, the
Plaintiff makes a takeitorleave it settlement offer of s>0.
• Defendant accepts or rejects. If Defendant rejects, Plaintiff has
to decide whether or not to go to trial at cost c>0 to the Plaintiff
and at cost d>0 to the Defendant.
• If the case goes to trial, Plaintiff wins the amount w with
the case goes to trial Plaintiff wins the amount with
probability p and loses (payoff=0) with probability 1p.
• Assume that pw < c, and this fact is common knowledgea
critical assumption. The Game in Extensive Form A Specific Parameterization of
th
the Game to Play
Pl
• k, cost of filing a
cost of filing
lawsuit = 10
• c=d=cost of a trial
=30
• Settlement off =50
• Winnings, w=100
• Probability the
Plaintiff wins,p=.10 The Lawsuit is Not Credible
The Lawsuit is Not Credible • The subgame
perfect
equilibrium is that
ilib
th
the Plaintiff does
nothing, as pw<c,
so that pw
so that pwkc <k How to Make the Nuisance
Lawsuit Credible?
• •
•
•
•
•
• Consider the strategic move by which Plaintiff prepays his lawyer the
the strategic move by which Plaintiff pre
his lawyer the
costs of a trial, c, upfront (Plaintiff puts money in a nonrefundable
retainer account guaranteeing the attorney’s future availability for
trial).
trial).
In this case the payoff to Give Up changes from –k to –kc.
Plaintiff goes to trial if pw>0, i.e. if there is any positive probability p
of winning the amount
of winning the amount w.
This further implies that Plaintiff prefers settlement to trial only if
s>pw.
Defendant is now willing to settle for any amount s<pw+d.
So the settlement range is [pw, pw+d]. If the Plaintiff can make a
takeitorleaveit offer, what is the settlement amount s?
Outcome is Plaintiff sues, offers to settle, and a settlement is reached. Further Considerations
Further Considerations
• If Plaintiff settles, her payoff is skc, where k is the cost of
filing the lawsuit (threat), c is the retainer cost.
• If Plaintiff can make a takeitorleaveit offer of s=pw+d,
then the question remains as to whether the nuissance
then the question remains as to whether the nuissance
lawsuit should be brought in the first place.
• Since doing nothing yields a payoff of 0 (this is a nuisance
lawsuit, no real harm done), then we must have that: skc>0
and if s=pw+d, then we require that pw+dkc>0.
• Can incorporate malice, whereby d enters positively in the
incorporate malice whereby enters positively in the
plaintiff’s payoff from going to trial Another Example: Criminal Law
•
• •
•
•
•
•
•
• Accused criminal can plead guilty or not.
If he pleads not guilty, the prosecutor can offer a full sentence, valued at s or a
reduced sentence valued at r. For the prosecutor, s>r, but for the accused, the
payoffs are losses:
payoffs are losses: s, r, so r>s; the accused strictly prefers a reduced sentence.
so r> the accused strictly prefers reduced sentence
Accused can accept prosecutor’s offer or reject it and go to trial.
Probability of being convicted, p, is assumed known.
Cost to the prosecutor of taking the case to trial=c.
to the prosecutor of taking the case to trial
Cost to the accused of defending himself at trial=d.
Suppose d/s > 1p, probability of an acquittal. In this case, in equilibrium the
prosecutor offers no reduction, and the accused pleads guilty and gets sentence s.
Strategic move: Accused requests a public defender (free lawyer), so that d=0.
In response, the prosecutor offers a reduced sentence, r=psc<s; Accused accepts.
Logic: By eliminating the cost to the accused of going to trial, the accused will
always go to trial rather than plead guilty since ps<s if p<1. Knowing this,
prosecutor will cut a deal offering a reduced sentence of r¥ psc, greater than or
equal to the prosecutor’s expected payoff from going to trial. The accused will
only accept the lowest possible reduced sentence r=
only accept the lowest possible reduced sentence, r=psc. Prosecutor agrees to
Prosecutor agrees to
this deal as he is made indifferent between accepting it and going to trial. Other Means of Acquiring
Credibility
• Delegation of your decision to a disinterested /nonemotional
third party.
– If you do not pay by this date, a repo man is scheduled to take your car. • Legally binding contracts.
bi
– If you violate parole, you will go back to prison. • Reputation may be a substitute for commitment in a repeated
eputat
be subst
co
game setting –more on this next week.
– You are known to get angry if you don’t get your way. So you get your
way
way. • Rational irrationality.
– You know I’m so crazy that I will follow through with the incredible
threat. Ways to Counter Strategic Moves
Ways to Counter Strategic Moves
• Counter sunk costs with sunk costs.
– In the nuisance lawsuit game the defendant can pay his lawyer in advance
for the cost of trial Better yet hire in house lawyers to deter nuisance
for the cost of a trial. Better yet, hire in house lawyers to deter nuisance
suits in the first place. • Eliminate communication/observation – if a strategic move is not
communicated/observed it is useless
communicated/observed, it is useless.
– For example, I’m not going to look at your sign that reads “homeless,
please help.” • Undermine an opponent’s need to carry out threats.
an opponent need to carry out threats
– I won’t tell if you make an exception for me; it can be “our little secret.” • Salami tactics: Reduce an opponent’s threat by complying to a very
small degree especially repeatedly over time like cutting off and
small degree, especially repeatedly over time, like cutting off and
offering thin slices of salami. In this way, the threat is never triggered,
though you may fail to completely comply.
– “Sorry for the delay. Times are tough for me right now. I can repay you this amount
now and will send the rest of what I owe you shortly.” ...
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This note was uploaded on 12/26/2011 for the course ECON 171 taught by Professor Charness,g during the Fall '08 term at UCSB.
 Fall '08
 Charness,G

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