CSULA Testing Mixed Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer Discussion

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Testing Mixed-Strategy Equilibria When Players Are
Heterogeneous: The Case of Penalty Kicks in Soccer
By P.-A. CHIAPPORI, S. LEVITT,
The concept of mixed strategy is a fundamental component of game theory, and its normative importance is undisputed. However, its
empirical relevance has sometimes been viewed
with skepticism. The main concern over the
practical usefulness of mixed strategies relates
to the “indifference” property of a mixedstrategy equilibrium. In order to be willing to
play a mixed strategy, an agent must be indifferent between each of the pure strategies that
are played with positive probability in the
mixed strategy, as well as any combination of
those strategies. Given that the agent is indifferent across these many strategies, there is no
beneŽ t to selecting precisely the strategy that
induces the opponent to be indifferent, as required for equilibrium. Why an agent would, in
the absence of communication between players,
choose exactly one particular randomization is
not clear.1
Of course, whether agents, in real life, actually play Nash equilibrium mixed strategies is
ultimately an empirical question. The evidence
to date on this issue is based almost exclusively
on laboratory experiments (e.g., Barry O’Neill,
1987; Amnon Rapoport and Richard B. Boebel,
1992; Dilip Mookherjee and Barry Sopher,
AND
T. GROSECLOSE*
1994; Jack Ochs, 1995; Kevin A. McCabe et al.,
2000). The results of these experiments are
mixed. O’Neill (1987) concludes that his experimental evidence is consistent with Nash mixed
strategies, but that conclusion was contested by
James N. Brown and Robert W. Rosenthal
(1990). With the exception of McCabe et al.
(2000), which looks at a three-person game, the
other papers generally reject the Nash mixedstrategy equilibrium.
While much has been learned in the laboratory, there are inherent limitations to such studies. It is sometimes argued that behavior in the
simpliŽ ed, artiŽ cial setting of games played in
such experiments need not mimic real-life behavior. In addition, even if individuals behave
in ways that are inconsistent with optimizing
behavior in the lab, market forces may discipline such behavior in the real world. Finally,
interpretation of experiments rely on the assumption that the subjects are maximizing the
monetary outcome of the game, whereas there
may be other preferences at work among subjects (e.g., attempting to avoid looking foolish)
that distort the results.2
Tests of mixed strategies in nonexperimental
data are quite scarce. In real life, the games
played are typically quite complex, with large
strategy spaces that are not fully speciŽ ed ex
ante. In addition, preferences of the actors may
not be perfectly known. We are aware of only
one paper in a similar spirit to our own research.
Using data from classic tennis matches, Mark
Walker and John Wooders (2001) test whether
the probability the player who serves the ball
wins the point is equal for serves to the right and
* Chiappori: Department of Economics, University of
Chicago, 1126 East 59th Street, Chicago, IL 60637; Levitt:
Department of Economics, University of Chicago; Groseclose: Graduate School of Business, Stanford University,
518 Memorial Way, Stanford, CA 94305. The paper was
presented at Games 2000 in Bilbao and at seminars in Paris
and Chicago. We thank D. Braun, J. M. Conterno, R.
Guesnerie, D. Heller, D. Mengual, P. Reny, B. Salanie?, and
especially J. L. Ettori for comments and suggestions, and M.
Mazzocco and F. Bos for excellent research assistance. Any
errors are ours.
1
The theoretical arguments given in defense of the concept of mixed-strategy equilibria relate either to puriŽ cation
(John C. Harsanyi, 1973), or to the minimax property of the
equilibrium strategy in zero-sum games. For recent elaborations on these ideas, see Authur J. Robson (1994) and Phil
Reny and Robson (2001).
2
The ultimatum game is one instance in which experimental play of subjects diverges substantially from the
predicted Nash equilibrium. Robert Slonim and Alvin E.
Roth (1998) demonstrate that raising the monetary payoffs
to experiment participants induces behavior closer to that
predicted by theory, although some disparity persists.
1138
VOL. 92 NO. 4
CHIAPPORI ET AL.: TESTING MIXED-STRATEGY EQUILIBRIA
left portion of the service box, as would be
predicted by theory. The results for tennis
serves is consistent with equilibrium play.3
In this paper, we study penalty kicks in soccer. This application is a natural one for the
study of mixed strategies. First, the structure of
the game is that of “matching pennies,” thus
there is a unique mixed-strategy equilibrium.
Two players (a kicker and a goalie) participate
in a zero-sum game with a well-identiŽ ed strategy space (the kicker’s possible actions can be
reasonably summarized as kicking to either the
right, middle, or left side of the goal; the goalie
can either jump to the right or left, or remain in
the middle). Second, there is little ambiguity to
the preferences of the participants: the kicker
wants to maximize the probability of a score
and the goalie wants to minimize scoring. Third,
enormous amounts of money are at stake, both
for the franchises and the individual participants. Fourth, data are readily available and are
being continually generated. Finally, the participants know a great deal about the past history
of behavior on the part of opponents, as this
information is routinely tracked by soccer clubs.
We approach the question as follows. We
begin by specifying a very general game in
which each player can take one of three possible
actions {left, middle, right}. We make mild
general assumptions on the structure of the payoff (i.e., scoring probabilities) matrix; e.g., we
suppose that scoring is more likely when the
goalie chooses the wrong side, or that rightfooted kickers are better when kicking to the
left.4 The model is tractable, yet rich enough to
generate complex and sometimes unexpected
predictions. The empirical testing of these predictions raises very interesting aggregation
problems. Strictly speaking, the payoff matrix is
match-speciŽ c (i.e., varies depending on the
identities of the goalie and the kicker). In our
3
Much less relevant to our research is the strand of
literature that builds and estimates game-theoretic models
that sometimes involve simultaneous-move games with
mixed-strategy equilibria such as Kenneth Hendricks and
Robert Porter (1988) and Timothy F. Bresnahan and Peter
C. Reiss (1990).
4
These general assumptions were suggested by common
sense and by our discussions with professional soccer players. They are testable and supported by the data.
1139
data, however, we rarely observe multiple observations for a given pair of players.5 This
raises a standard aggregation problem. While
the theoretical predictions hold for any particular matrix, they may not be robust to aggregation; i.e., they may fail to hold on average for an
heterogeneous population of games. We investigate this issue with some care. We show that
several implications of the model are preserved
by aggregation, hence can be directly taken to
data. However, other basic predictions (e.g.,
equality of scoring probabilities across right,
left, and center) do not survive aggregation in
the presence of heterogeneity in the most general case. We then proceed to introduce additional assumptions into the model that provide a
greater range of testable hypotheses. Again,
these additional assumptions, motivated by the
discussions with professional soccer players,
are testable and cannot be rejected in the data.
The assumptions and predictions of the
model are tested using a data set that includes
virtually every penalty kick occurring in the
French and Italian elite leagues over a period of
three years—a total of 459 kicks. A critical
assumption of the model is that the goalie and
the kicker play simultaneously. We cannot reject this assumption empirically; the direction a
goalie or kicker chooses on the current kick
does not appear to in uence the action played
by the opponent. In contrast, the strategy chosen
by a goalie today does depend on a kicker’s past
history. Kickers, on the other hand, play as if all
goalies are identical. We also Ž nd that all the
theoretical predictions that are robust to aggregation (hence that can be tested directly on the
total sample) are satisŽ ed. Finally, using the
result that goalies appear to be identical, we test,
and do not reject, the null hypothesis that scoring probabilities are equal for kickers across
right, left, and center. Also, subject to the limitations that aggregation imposes on testing
goalie behavior, we cannot reject equal scoring
probabilities with respect to goalies jumping
right or left (goalies almost never stay in the
5
Even for a given match, the matrix of scoring probabilities may moreover be affected by the circumstances of
the kick. We Ž nd, for instance, that scoring probabilities
decline toward the end of the game.
1140
THE AMERICAN ECONOMIC REVIEW
middle). It is important to note, however, that
some of our tests have relatively low power.
The remainder of the paper is structured as
follows. Section I develops the basic model.
Section II analyzes the complexities that arise in
testing basic hypotheses in the presence of heterogeneity across kickers and goalies. We note
which hypotheses are testable when the researcher has only a limited number of kicks per
goalie–kicker pair, and we introduce and test
restrictions on the model that lead to a richer set
of testable hypotheses given the limitations of
the data. Section III presents the empirical tests
of the predictions of the model. Section IV
concludes.
I. The Framework
A. Penalty Kicks in Soccer
According to the rule, “a penalty kick is
awarded against a team which commits one of
the ten offenses for which a direct free kick is
awarded, inside its own penalty area and while
the ball is in play.”6 The maximum speed the
ball can reach exceeds 125 mph. At this speed,
the ball enters the goal about two-tenths of a
second after having been kicked. This means
that a keeper who jumps after the ball has been
kicked cannot possibly stop the shot (unless it is
aimed at him). Thus the goalkeeper must choose
the side of the jump before he knows exactly
where the kick is aimed.7 It is generally believed that the kicker must also decide on the
side of his kick before he can see the keeper
move. A goal may be scored directly from a
penalty kick, and it is actually scored in about
6
The ball is placed on the penalty mark, located 11 m
(12 yds) away from the midpoint between the goalposts.
The defending goalkeeper remains on his goal line, facing
the kicker, between the goalposts until the ball has been
kicked. The players other than the kicker and the goalie are
located outside the penalty area, at least 9.15 m (10 yds)
from the penalty mark; they cannot interfere in the kick.
7
According to a former rule, the goalkeeper was not
allowed to move before the ball was hit. This rule was never
enforced; in practice, keepers always started to move before
the kick. The rule was modiŽ ed several years ago. According to the new rule, the keeper is not allowed to move
forward before the ball is kicked, but he is free to move
laterally.
SEPTEMBER 2002
four kicks out of Ž ve.8 Given the amounts of
money at stake, the value of any factor affecting
the outcome even slightly is large.
In all Ž rst-league teams, goalkeepers are especially trained to save penalty kicks, and the
goalie’s trainer keeps a record of the kicking
habits of the other teams’ usual kickers. Conventional wisdom suggests that a right-footed
kicker (about 85 percent of the population) will
Ž nd it easier to kick to his left (his “natural
side”) than his right; and vice versa for a leftfooted kicker. The data strongly support this
claim, as will be demonstrated. Thus, throughout the paper we focus on the distinction
between the “natural” side (i.e., left for a rightfooted player, right for a left-footed player) and
the “nonnatural” one. We adopt this convention
in the remainder. For the sake of readability,
however, we use the terms “right” and “left”
in the text, although technically these would
be reversed for (the minority of) left-footed
kickers.
B. The Model
Consider a large population of goalies and
kickers. At each penalty kick, one goalie and
one kicker are randomly matched. The kicker
(respectively, the goalie) tries to maximize
(minimize) the probability of scoring. The
kicker may choose to kick to (his) right, his left,
or the center of the goal. Similarly, the goalie
may choose to jump to (the kicker’s) left, right,
or to remain at the center. When the kicker and
the goalie choose the same side S (S 5 R, L),
the goal is scored with some probability P S. If
the kicker chooses S (S 5 R, L) while the
goalie either chooses the wrong side or remains
at the center, the goal is scored with probability
p S . P S. Here, 1 2 p S can be interpreted as
the probability that the kick goes out or hits the
post or the bar; the inequality p S . P S re ects
the fact that when the goalie makes the correct
choice, not only can the kick go out, but in
addition it can be saved. Finally, a kick at the
8
The average number of goals scored per game slightly
exceeds two on each side. About one-half of the games end
up tied or with a one-goal difference in scores. In these
cases, the outcome of a penalty kick has a direct impact on
the Ž nal outcome.
VOL. 92 NO. 4
CHIAPPORI ET AL.: TESTING MIXED-STRATEGY EQUILIBRIA
center is scored with probability m when the
goalie jumps to one side, and is always saved if
the goalie stays in the middle. Technically, the
kicker and the goalie play a zero-sum game.
Each strategy space is {R, C, L}; the payoff
matrix is given by:
TABLE 1—OBSERVED SCORING PROBABILITIES,
BY FOOT AND SIDE
Goalie
Ki
L
C
R
L
C
R
PL
m
pR
pL
0
pR
pL
m
PR .
It should be stressed that, in full generality, this
matrix is match-speciŽ c. The population is
characterized by some distribution d f (P R, P L,
pR, pL, m) of the relevant parameters. We assume that the speciŽ c game matrix at stake is
known by both players before the kick; this is a
testable assumption, and we shall see it is not
rejected by the data. Finally, we assume both
players move simultaneously. Again, this assumption is testable and not rejected.
We now introduce three assumptions on scoring probabilities, that are satisŽ ed by all
matches. These assumptions were suggested to
us by the professional goalkeepers we talked to,
and seem to be unanimously accepted in the
profession.
ASSUMPTION SC (“Sides and Center”):
(SC)
p R . PL
pL . PR
(SC9)
pR . m
pL . m.
ASSUMPTION NS (“Natural Side”):
PL $ PR .
(NS)
pL $ pR
ASSUMPTION KS (“Kicker’s Side”):
(KS)
pR 2 P R $ pL 2 PL .
Assumption (SC) states Ž rst that, if the kicker
knew with certainty which direction the goalie
would jump, he would choose to kick to the
other direction [relation (SC)]. Also, if the
goalie jumps to the kicker’s left (say), the scoring probability is higher for a kick to the right
than to the center [relation (SC)]. The natural
side (NS) assumption requires that the kicker
kicks better on his natural side, whether the
keeper guesses the side correctly or not. Finally,
Correct
side
Middle or
wrong side
63.6 percent
43.7 percent
94.4 percent
89.3 percent
Kicker
Natural side (“left”)
Opposite side (“right”)
Gi
1141
(KS) states that not only are kicks to the natural
side less likely to go out, but they are also less
easy to save.9
These assumptions are fully supported by the
data, as it is clear from Table 1. The scoring
probability when the goalie is mistaken varies
between 89 percent and 95 percent (depending
on the kicking foot and the side of the kick),
whereas it ranges between 43 percent and 64
percent when the goalkeeper makes the correct
choice, substantiating relation (SC). Also, the
scoring probability is always higher on the kicker’s natural side (Assumption NS), and the difference is larger when the goalie makes the
correct choice (Assumption KS). Regarding
(SC9), our data indicate that the scoring probability, conditional on the goalie making the
wrong choice, is 92 percent for a kick to one
side versus 84 percent for a kick in the middle.10
C. Equilibrium: A First Characterization
The game belongs to the “matching penny”
family. It has no pure-strategy equilibrium, but
9
If the goalie makes the wrong choice, the kicker scores
unless the kick is out, which, for side X (X 5 L, R),
happens with probability 1 2 p X. If the goalie guesses the
correct side, failing to score means either that the kick is out
(which, because of independence, occurs again with probability 1 2 p X), or that the kick is saved. Calling s X the
latter probability, one can see that
P X 5 p X 2 sX
so that (KS9) is equivalent to
sR $ sL.
10
These results should however be interpreted with caution, since aggregation problems may arise (see below).
1142
THE AMERICAN ECONOMIC REVIEW
it always admits a unique mixed-strategy equilibrium, as stated in our Ž rst proposition.
PROPOSITION 1: There exist a unique mixedstrategy equilibrium. If
~C m !
m#
p LpR 2 P LP R
pR 1 pL 2 PL 2 PR
then both players randomize over {L, R} (“restricted randomization”). Otherwise both players randomize over {L, C, R} (“general
randomization”).
The proof relies on straightforward (although
tedious) calculations. The interested reader is
referred to Chiappori et al. (2000).
In a restricted randomization (RR) equilibrium, the kicker never chooses to kick at the
center, and the goalie never remains in the center. An equilibrium of this type obtains when
the probability m of scoring when kicking at the
center is small enough. The scoring probability
is identical for both sides:
Pr~scorezS 5 L! 5 Pr~scorezS 5 R!
5
p LpR 2 P LP R
pL 1 p R 2 P L 2 P R
whereas a kick in the middle scores with strictly
smaller probability m.11 In a generalized randomization (GR) equilibrium, on the other
hand, both the goalie and the kicker choose
right, left, or in the middle with positive probability, and the equilibrium scoring probabilities
are equal.
Thus, kickers do not kick to the center unless
the scoring is large enough, whereas they always kick to the sides with positive probability.
With heterogeneous matches, this creates a selection bias, with the consequence that the aggregate scoring probability (i.e., proportion to
kicks actually scored) should be larger for kicks
to the center. We shall see that this pattern is
actually observed in our data.
11
Also, if pR 5 pL, the goalie and the kicker play the
same mixed strategy.
SEPTEMBER 2002
D. Properties of the Equilibrium
We now present several properties of the
equilibrium that will be crucial in deŽ ning our
empirical tests.
PROPOSITION 2: At the unique equilibrium
of the game, the following properties hold true:
1. The kicker’s and the goalie’s randomization
are independent.
2. The scoring probability is the same
whether the kicker kicks right, left, or center whenever he does kick at the center
with positive probability. Similarly, the
scoring probability is the same whether the
goalie jumps right, left, or center whenever he does remain at the center with
positive probability.
3. Under Assumption (SC), the kicker is always
more likely to choose C than the goalie.
4. Under Assumption (SC), the kicker always
chooses his natural side less often than the
goalie.
5. Under Assumptions (SC) and (NS), the
keeper chooses the kicker’s natural side L
more often that the opposite side R.
6. Under Assumptions (SC) and (KS), the
kicker chooses his natural side L more often
that the opposite side R.
7. Under Assumptions (SC), (NS), and (KS), the
pattern (L, L) (i.e., the kicker chooses L and
the goalie chooses L) is more likely than
both (L, R) and (R, L), which in turn are both
more likely than (R, R).
Properties 1 and 2 are standard characterizations of a mixed-strategy equilibrium. Properties 3 and 4 are direct consequences of the form
of the matrix and of Assumption (SC), and
provide wonderful illustrations of the logic of
mixed-strategy equilibria. For instance, the
kicker’s probability of kicking to the center
must make the goalie indifferent between jumping or staying (and vice versa for the goalie).
Now, kicking at the center when the keeper
stays is very damaging for the kicker (the scoring probability is zero), so it must be the case
that at equilibrium this situation is very rare (the
goalie should stay very rarely). Conversely,
from the goalie’s perspective, kicks to the cen-
VOL. 92 NO. 4
CHIAPPORI ET AL.: TESTING MIXED-STRATEGY EQUILIBRIA
ter are not too bad, even if he jumps [they are
actually better than kicks to the opposite side by
(SC)], hence their equilibrium probability is
larger.
Finally, the same type of reasoning applies
the statements 5, 6, and 7. Assume the goalie
randomizes between R and L with equal conditional probabilities. By Assumption (NS), the
kicker would then be strictly better off kicking
L, a violation of the indifference condition;
hence at equilibrium the goalie should choose L
more often. Similarly, Assumption (KS) implies
that, should the kicker randomize equally between L and R, jumping to the right would be
more effective from the goalie’s viewpoint.
Again, indifference requires more frequent
kicks to the left. In all cases, the key remark is
that the kicker’s scoring probabilities are relevant for the keeper’s strategy (and conversely),
a conclusion that is typical of mixed-strategy
equilibria, and sharply contrasts with standard
intuition.
II. Heterogeneity and Aggregation
The previous propositions apply to any
particular match. However, match-speciŽ c
probabilities are not observable; only populationwide averages are. With a homogeneous
population (i.e., assuming that the game matrix is identical across matches) this would
not be a problem, since populationwide averages exactly re ect probabilities. Homogeneity, however, is a very restrictive assumption,
that does not Ž t the data well (as demonstrated below). Heterogeneity will arise if
players have varying abilities or characteristics, and may even be affected by the environment (time of the game, Ž eld condition,
stress, fatigue, etc.). Then, a natural question
is: which of the predictions above are preserved by aggregation, even in the presence of
some arbitrary heterogeneity?
The following result summarizes the predictions of the model that are preserved by
aggregation:
PROPOSITION 3: For any distribution df(PR,
P L, pR, p L, m ), the following hold true, under
Assumption (SC):
1143
(i) The total number of kicks to the center is
larger than the total number of kicks for
which the goalie remains at the center.
(ii) The total number of kicks to the kicker’s
left is smaller than the total number of
jumps to the (kicker’s) left.
(iii) If Assumption (NS) is satisŽ ed for all
matches, then the number of jumps to the
left is larger than the number of jumps to
the right.
(iv) If Assumption (KS) is satisŽ ed for all
matches, then the number of kicks to the
left is larger than the number of kicks to
the right.
(v) If Assumptions (NS) and (KS) are satisŽ ed
for all matches, then the pattern (L, L) (i.e.,
the kicker chooses L and the goalie
chooses L) is more frequent than both (L,
R) and (R, L), which in turn are both more
frequent than (R, R).
Other results, however, may hold for each
match but fail to be robust to aggregation. For
instance, the prediction that the scoring probability should be the same on each side does not
hold on aggregate, even when it works for each
possible match. Assume, for instance, that there
are two types of players, who differ in ability
and equilibrium side, say, the best players shoot
relatively more often to the left at equilibrium.
Then a left kick is more likely to come from a
stronger player and therefore has a higher
chance of scoring. Econometrically, this is
equivalent to stating that a selection bias arises
whenever the side of the kick is correlated with
the scoring probabilities; and theory asserts it
must be, since it is endogenously determined by
the probability matrix.
The heterogeneity problem may arise even
when the same kicker and goalie are matched
repeatedly, since scoring probabilities are
affected by various exogenous variables.12
Therefore, the equal scoring probability property should not be tested on raw data, but
instead conditional on observables.13 However,
12
For instance, we Ž nd that the scoring probability is
larger for a penalty kick during the Ž rst 15 minutes of the
game, and smaller for the last half hour.
13
We Ž nd, however, that while scoring probabilities do
change over time during the game, the probabilities of
1144
THE AMERICAN ECONOMIC REVIEW
conditioning on covariates is not enough.
While the total number of kicks available is
fairly large, they mostly represent different
pairings of kickers and goalies. For any given
pairing, there are at most three kicks, and
often one or two (or zero). Match-speciŽ c
predictions are thus very difŽ cult to test. Two
solutions exist at this point. First, it is possible to test the predictions that are preserved
by aggregation. Second, speciŽ c assumptions
on the form of the distribution will allow
testing of a greater number of predictions.
Of course, it is critical that these assumptions be testable and not rejected by the
data. In what follows, we use the following
assumption:
ASSUMPTION IG (Identical Goalkeepers):
For any match between a kicker i and a goalie
j, the parameters P R, P L, p R, p L, and m do not
depend on j.
SEPTEMBER 2002
ity, the corresponding scoring probability
is the same as when kicking at either side,
irrespective of the goalkeeper.
(iv) The scoring probability is the same
whether the goalkeeper jumps right or left,
irrespective of the goalkeeper. If the kicker
kicks at the center with positive probability, the corresponding scoring probability
is the same as when kicking at either side,
irrespective of the goalkeeper.
(v) Conditional on not kicking at the center,
the kicker always chooses his natural side
less often than the goalie.
From an empirical viewpoint, Assumption
(IG) has a key consequence: all the theoretical
results, including those that are not preserved by
aggregation, can be tested kicker by kicker,
using all kicks by the same kicker as independent draws of the same game.
III. Empirical Tests
In other words, while kickers differ from
each other, goalies are essentially identical. The game matrix is kicker-speciŽ c, but it
does not depend on the goalkeeper; for a
given kicker, each kicker– goalie pair faces
the same matrix whatever the particular
goalie involved.
Note, Ž rst, that this assumption can readily be
tested; as we shall see, it is not rejected by the
data. Also, Assumption IG, if it holds true, has
various empirical consequences.
PROPOSITION 4: Under Assumption IG, for
any particular kicker i, the following hold true:
(i) The kicker’s strategy does not depend on
the goalkeeper.
(ii) The goalkeeper’s strategy is identical for
all goalkeepers.
(iii) The scoring probability is the same
whether the kicker kicks right or left, irrespective of the goalkeeper. If the kicker
kicks at the center with positive probabil-
kicking to the right or to the left are not signiŽ cantly
affected. This suggests that the bias induced by aggregation
over games with different covariates may not be too severe.
We test the assumptions and predictions of
the model in the previous sections using a
data set of 459 penalty kicks. These kicks
encompass virtually every penalty kick taken
in the French Ž rst league over a two-year
period and in the Italian Ž rst league over a
three-year period. The data set was assembled
by watching videotape of game highlight
Ž lms. For each kick, we know the identities of
the kicker and goalie, the action taken by both
kicker and goalie (i.e., right, left, or center),
which foot the kicker used for the shot, and
information about the game situation such as
the current score, minute of the game, and the
home team. A total of 162 kickers and 88
goalies appear in the data. As a consequence
of the relatively small number of observations
in the data set, some of our estimates are
imprecise, leading our tests to have relatively
low power to discriminate between competing
hypotheses. Because the power of some of
the tests of the model increases with the number of observations per kicker, in some cases
we limit the sample to either the 41 kickers
with at least four shots (58 percent of the
total observations) or the nine kickers with
at least eight shots (22 percent of the total
observations).
VOL. 92 NO. 4
CHIAPPORI ET AL.: TESTING MIXED-STRATEGY EQUILIBRIA
A. Testing the Assumption That Kickers and
Goalies Move Simultaneously
Before examining the predictions of the
model, we Ž rst test the fundamental assumption
of the model: the kicker and goalie move simultaneously. Our proposed test of this assumption
is as follows. If the two players move simultaneously, then conditional on the player’s and
the opponent’s past history, the action chosen
by the opponent on this penalty kick should not
predict the other player’s action on this penalty
kick. Only if one player moves Ž rst (violating
the assumption of a simultaneous-move game)
should the other player be able to tailor his
action to the opponent’s actual choice on this
particular kick. We implement this test in a
linear probability regression of the following
form:14
(SM)
R Ki 5 Xia 1 b R Gi 1 g R# Ki 1 d R# Gi 1 « i
G
where R K
i (respectively, R i ) is a dummy for
whether, in observation i, the kicker shoots
#G
(keeper jumps) right, R# K
i (R i ) is the proportion
of kicks by the kicker (of jumps by the goalie)
going right on all shots except this one,15 and X
is a vector of covariates that includes a set of
controls for the particulars of the game situation
at the time of the penalty kick: Ž ve indicators
corresponding to the minute of the game in
which the shot occurs, whether the kicker is on
the home team, controls for the score of the
game immediately prior to the penalty kick,
and interaction terms that absorb any systematic differences in outcomes across leagues or
across years within a league. The key parameter in this speciŽ cation is b , the coefŽ cient
on whether the goalie jumps right on this
kick. In a simultaneous move game, b should
be equal to zero.
Results from the estimation of equation
(SM) are presented in Table 2. The odd-numbered columns include all kickers; the even
14
Probit regressions give similar results, although the
interpretation of the coefŽ cients is less straightforward.
15
Similar tests have been run using only penalty kicks
prior to the one at stake. As in Table 2, we are unable to
reject the null hypothesis of simultaneous moves.
1145
columns include only kickers with at least
four penalty kicks in the sample. Kickers with
few kicks may not have well-developed reputations as to their choice of strategies.16 Columns 1 and 2 include only controls for the
observed kicker and goalie behaviors. Columns 3 and 4 add in the full set of covariates
related to the particulars of the game situation
at the time of the pe