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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

bene 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 ONeill,

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. ONeill (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

simpli ed, arti 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 speci 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 puri 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-identi ed strategy space (the kickers 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-speci 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 yearsa 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 kickers past

history. Kickers, on the other hand, play as if all

goalies are identical. We also nd that all the

theoretical predictions that are robust to aggregation (hence that can be tested directly on the

total sample) are satis 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 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

goaliekicker 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 modi 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 ve.8 Given the amounts of

money at stake, the value of any factor affecting

the outcome even slightly is large.

In all rst-league teams, goalkeepers are especially trained to save penalty kicks, and the

goalies 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

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 kickers) 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 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 1OBSERVED 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-speci 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 speci 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 satis 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 (Kickers Side):

(KS)

pR 2 P R $ pL 2 PL .

Assumption (SC) states 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 kickers 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 kickers 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 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 de ning our

empirical tests.

PROPOSITION 2: At the unique equilibrium

of the game, the following properties hold true:

1. The kickers and the goalies 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 kickers 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

kickers 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 goalies 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 goalies viewpoint.

Again, indifference requires more frequent

kicks to the left. In all cases, the key remark is

that the kickers scoring probabilities are relevant for the keepers 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-speci 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 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, 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 kickers

left is smaller than the total number of

jumps to the (kickers) left.

(iii) If Assumption (NS) is satis 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 satis 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 satis 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 nd that the scoring probability is

larger for a penalty kick during the rst 15 minutes of the

game, and smaller for the last half hour.

13

We 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-speci c

predictions are thus very dif cult to test. Two

solutions exist at this point. First, it is possible to test the predictions that are preserved

by aggregation. Second, speci 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-speci 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, 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 kickers strategy does not depend on

the goalkeeper.

(ii) The goalkeepers 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 signi 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 rst league over a two-year

period and in the Italian rst league over a

three-year period. The data set was assembled

by watching videotape of game highlight

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 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 players and

the opponents past history, the action chosen

by the opponent on this penalty kick should not

predict the other players action on this penalty

kick. Only if one player moves rst (violating

the assumption of a simultaneous-move game)

should the other player be able to tailor his

action to the opponents 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: 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 speci cation is b , the coef 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 coef 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