Do Professional Football Players Follow the Optimal Strategies

in Penalty Shootout?

Takaya Koizumi, Ryohei Orihara, Yuichi Sei, Yasuyuki Tahara and Akihiko Ohsuga

Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo, Japan

Keywords:

Penalty Shootout, Optimal Strategy, Big Data.

Abstract:

Do people and companies choose optimal strategies under various situations? If not so, why not? Pursuing

the reason for this helps to understand individuals and companies. In addition, game theory has been heavily

involved in the understanding of sports, economics and other social sciences. In this study, we focused on

football’s penalty shootout where data is relatively easy to collect, mixed strategy can be applied and making

a pay-off matrix considering our own probability is possible. Using the pay-off matrix, we obtained optimal

strategy of the kicker in the penalty shootout and revealed the gap between the optimal strategy and actual

action taken by players. We compared the probability distribution for each data attribute in the dataset in

order to obtain the cause of the gap. We use 100 professional penalty shootouts (total 1032 kicks) which

were collected from internet video site during the period from 2001-2016. Experimental results showed that

there was a gap between the optimal strategy and the actual action taken by players and that it also suggested

the position and team attributes and temporary scores of the shootout and kicking order involved in the gap.

Considering them we made the hypothesis and estimated the cause of the gap. We hope this method can apply

to other ﬁelds than sports.

1 INTRODUCTION

‘Optimal strategy’ is deﬁned the strategy which is

played to maximize their own proﬁt by the subject

without cooperation with others. The study has been

done to ﬁnd optimal strategies in various ﬁelds such as

optimal investment behavior (Anderson et al., 2008)

in the economic ﬁelds and optimal strategy of serve

in tennis (Gale, 1971) in sports ﬁelds.

However, it is another question whether the op-

timal strategy matches the actual strategy taken by

players. Because even if people and companies are

aware of the optimal strategy for the things, they have

the possibility not to follow the strategy because of

the environmental or psychological factors. In other

words, there is a gap between optimal strategy and

actual strategy taken by players. Identifying the cause

of the gap helps to understand the psychological state

in the decision making of people and companies.

This study deals with football’s penalty shootout.

The penalty shootout takes place when there is no

score difference after the speciﬁed game time has

gone. Five kickers from each team alternately play

penalty kick(IFBA, 2016) (hereinafter, this is called

‘PK’) and compete for their number of success. Only

when it can not be determined at the end of 10 people,

they take the sudden death system.

The reasons for focusing on football PK game in

this paper are as follows.

• Relatively easy to collect data.

• Since player’s decision is made before the kicker’s

kick (Peiyong and Inomata, 2012), we can play

game theoretic approach using mixed strategy.

• In the penalty shootout, the optimal strategy may

differ from the actual strategy taken by players

because the environmental or psychological inﬂu-

ence greatly affects the performance (Jordet et al.,

2007).

The ﬂow of this study is divided into three steps:

Firstly we got an optimal strategy in the penalty

shootout. We created a goalkeeper (hereinafter, this is

called ‘GK’)’s pay-off matrix considering failure rate

f k

, f g

, and clariﬁed the kicker’s optimal strategy by

using Minimax. Next, we examined there was a gap

between the actual data and the optimal strategy. Fi-

nally, we extracted the data attributes which are con-

sidered to affect the gap. By referring this attribute,

we made the hypothesis and estimated the cause of

the gap.

454

Koizumi, T., Orihara, R., Sei, Y., Tahara, Y. and Ohsuga, A.

Do Professional Football Players Follow the Optimal Strategies in Penalty Shootout?.

DOI: 10.5220/0006591704540461

In Proceedings of the 10th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2018) - Volume 2, pages 454-461

ISBN: 978-989-758-275-2

2 KICKER’S OPTIMAL STRATEGY

In this section, we describe how to make GK’s pay-off

matrix and to get kicker’s optimal strategy.

2.1 GK’s Pay-off Matrix

In this paper, we consider the direction of the player’s

strategy in the penalty shootout to three directions :

left, center, right, which tear the goal into three equal

parts as seen from the kicker like Fig.1. Let π

i j

be the

GK’s pay-off. i and j represent kicker’s strategy and

GK’s strategy each other. i,j∈{L (left),C (center),R

(right)} is player’s strategy.

Figure 1: Deﬁnition of direction.

Now, we consider that GK’s pay-off is ‘1’ when

GK’s action matches the kicker’s action because GK

can block the score, and GK’s pay-off is ‘0’ when

GK’s action is different from kicker’s action because

GK can not block the score. Then GK’s pay-off is π

i j

=1 (i=j) and π

i j

=0 (i6=j). However, this GK’s pay-off

ignores the situation that GK fails to block the score

even if GK’s action matches the kicker’s action and

kicker fails to score even if GK’s action is different

from kicker’s action.

In this study, we use the failure rate f g

, f k

.Let

be the number of i=j and n

be the number of situ-

ation that GK can not block score even if GK’s action

matches the kicker’s action (i=j). Similarly, Let N

the number of i6=j and n

be the number of situation

that kicker can not score even if GK’s action is differ-

ent from the kicker’s action (i6=j). At this time, failure

rate f g

, f k

are deﬁned as follows.

f g

(1)

f k

(2)

A pay-off matrix considering the failure rate is

shown in Table 1.

2.2 Kicker’s Optimal Strategy

By using GK’s pay-off matrix, we calculate kicker’s

optimal strategy. First, we calculate expected value µ

Table 1: GK’s pay-off matrix considered failure rate.

i \j L C R

L 1- f g

f k

C f k

1- f g

f k

R f k

f k

1 − f g

of the pay-off in order to get the GK’s optimal strat-

egy. Let p

be the probability that the kicker choose

the action i. Then we calculate the expected value µ

of the pay-off at each of j∈ L, R, CWe compare mu

in each j = L, R, C and make the strategy with the

highest pay-off the GK’s optimal strategy. In addi-

tion, We also visualized this by mapping to the three-

dimensional space of p

on the x axis, p

on the y

axis and µ

on the z axis. Let this three-dimensional

space be .

By applying the Minimax method to GK’s optimal

strategy, we got the kicker’s optimal strategy. The

Minimax is a method by which a subject selects a

strategy so that the opponent’s pay-off is minimized.

3 THE GAP

Firstly, we found the center of gravity of the kicker’s

optimal strategy in space in order to obtain the gap

between the optimal strategy and actual strategy taken

by players. After that, we calculate the gap by com-

paring the probability distribution of the comparison

object with the center of gravity.

3.1 Center of Gravity

In this section, we obtain the center of gravity of the

kicker’s optimal strategy on the GK’s optimal strat-

egy. By ﬁnding the center of gravity of the area with

the smallest pay-off of the GK, we can reveal the op-

timal strategy within . Firstly, the GK’s optimal strat-

egy which is expressed in three-dimensional space is

projected into a two dimensional plane of p

on the

x-axis and p

on the y-axis. In order to calculate the

center of gravity a grid of 0.01 × 0.01 is placed on the

area corresponds to GK’s optimal strategy. The point

(u,v) is the center of gravity of the area if and only if:

(1) the number of grids for x < u is equal to the

number of grids for x > u, and

(2) the number of grids for y < v is equal to the

number of grids for y > v.

3.2 Calculating the Gap

Now, we describe how to ﬁnd a gap between the opti-

mal strategy and the actual strategy taken by players.

Do Professional Football Players Follow the Optimal Strategies in Penalty Shootout?

455

Let a

be a number of the kicker who satisﬁes a condi-

tion z (eg defender) and a

be the number of the kick-

ers who choose the action i and satisfy z. Then, the

condition z is represented by the following coordinate

in the X-Y plane of the :

(

) (3)

In addition, let d be the distance between center of

gravity and (

). The d represents the gap from

the optimal strategy and is used as an evaluation met-

ric in this study.

4 EXAMINATION AND RESULTS

4.1 Dataset

The dataset used is 100 professional penalty shootouts

(total 1032 kicks) which are collected from internet

video sites

during the period from 2001-2016. The

data includes 17 attributes, speciﬁcally date, num-

ber of spectators, team name, opponent team name,

kicking order, choice of the ﬁrst kick or the second

kick

, the country where the game took place, direc-

tion kicked, direction to which GK ﬂew, PK kicker

information (player name, position, nationality, dom-

inant foot), and GK information (player name, nation-

ality, height).

4.2 Calculating GK’s Pay-off Matrix

We make a GK’s pay-off matrix by the method ex-

plained in 2.1. In this study, the probabilistic model is

created for each of the dominant feet of the PK kick-

ers. In previous work, varying psychological charac-

ters were observed among right-footed and left-footed

players (Dane and S¸ekertekin, 2005), and statistically,

it was identiﬁed that there is a clear difference in

the strategic decision of penalty shootout (Palacios-

Huerta, 2003). Actually, different trends were also

seen in the preliminary experiments in the dataset of

this study, thus we consider that it is natural to divide

probability models with respect to the dominant foot.

Firstly, we calculate failure rate f g

, f k

in each

dominant foot. The result is shown in Table 2. The

GK’s pay-off matrix for right-footed using this is

shown in Table 3, for left-footed is shown in Table

https://www.youtube.com/

The referee tosses a coin and the team whose captain

wins the toss decides whether to take the ﬁrst or the second

kick. (IFBA, 2016)

Table 2: Failure Rate.

f g

f k

right 0.523 0.610 0 0.097 0.070 0.170

left 0.575 0440 0 0.121 0.018 0.342

Table 3: GK’s pay-off matrix (right-footed).

i \j L C R

L 0.391 0.070 0.070

C 0.170 1 0.017

R 0.097 0.097 0.477

Table 4: GK’s pay-off matrix (left-footed).

i \j L C R

L 0.561 0.018 0.018

C 0.342 1 0.342

R 0.121 0.121 0.425

4.3 Calculating Kicker’s Optimal

Strategy and Center of Gravity

We calculate the kicker’s optimal strategy and center

of gravity from the results in Section 4.2. Fig.2 and

Fig.3 show the kicker’s optimal strategy and center

of gravity for each dominant foot respectively, calcu-

lated by the method described in Section 2.2 and 3.1.

The x axis represents the probability of kicker kick-

ing to the right p

. The y axis represents the proba-

bility of kicker kicking to the left p

, and the z axis

is GK’s expected pay-off µ

. We get the kicker’s opti-

mal strategy by applying these GK’s optimal strategy

to Minimax.

From Fig.2 and Fig.3, we can see that the closer

and p

get to 0, the more GK’s expected pay-off

gets for both right-footed and left-footed. From this

result, it is found that the kicker should avoid to aim

to the center, and on the contrary, GK’s the pay-off is

low in the area µ

: 0.25-0.3 in each dominant foot,

thus this area is considered kicker’s optimal strategy.

In order to know the more accurate optimal strat-

egy, we obtain the center of gravity in the pay-off area

of µ

: 0.25-0.3. By determining the center of gravity

it becomes possible to represent the kicker’s optimal

strategy by one coordinate. As a result, the center of

gravity is (0.51, 0.41) and (0.33, 0.60) respectively in

right-footed left-footed as visualized by cross mark.

There is also a difference depending on the dominant

foot. We ﬁnd that right-footed and left-footed kicker’s

optimal strategy is right and left, respectively, which

means that the optimal strategy is different depending

on the dominant foot.

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

456

Figure 2: Center of gravity in kicker’s optimal strategy area

(right-footed).

Figure 3: Center of gravity in kicker’s optimal strategy area

(left-footed).

4.4 Calculating the Gap

Here, we verify whether the actual professional foot-

ball player’s penalty shootout follow the optimal strat-

egy. After that, we examine the probability distribu-

tion in each attribute and search for the cause of the

gap.

4.4.1 Comparing with Actual Data

As a result of examining the probability distribution

for each dominant foot in the data set, right-footed

player’s probability distribution is (p

, p

) = (0.37,

0.48) and left-footed player’s probability distribution

is (p

, p

) = (0.44, 0.40). The distance d from the

center of gravity was d = 0.162 in the case of right-

footed kickers and d = 0.230 in the case of left-footed

kickers. They also did not ﬁt within the area of the ex-

pected pay-off of 0.25 - 0.3 which is regarded as the

optimal strategy area, therefore it is found that pro-

fessional football players do not follow the optimal

strategy.

4.4.2 The Cause of the Gap

In Section 4.5.1, we found that the kicker’s optimal

strategy derived by the Minimax is different from

the behavior of professional football players. There-

fore we have a question why the professional athletes

tends to depart from a strategy that follows the opti-

mal strategy. In this section, we hypothesize the rea-

son.

First, we calculate the probability distribution (p

) for each attribute in the data set and got the gap d

from the center of gravity. Next, the attributes respon-

sible for the gap are identiﬁed based on the difference

between the gap d of the actual data obtained in Sec-

tion 4.5.1 and the gap d

. The results are shown in the

Table 5. The following nine attributes are analyzed :

position, nationality, the country where the game took

place, players in the side taking the ﬁrst/second kick,

temporary scores of the shootout difference, team at-

tributes, kicking order, the number of audiences, and

GK’s height. The score difference represents the su-

periority or inferiority at the moment when kicking

carries out. Nationality is classiﬁed into four regions

: Europe, Africa, East Asia, and South America be-

cause the amount of data for each nationality is too

small. For the same reason, we regard 6th, 7th and

8th kickers as one case in the kicking order.

When d −d

> 0, it indicates that it is closer to the

optimal strategy than the actual data, and conversely

when it is d − d

< 0, it indicates that it is farther.

The Table 5 shows d − d

for each attribute. We

consider that the cases with d − d

> 0 for each dom-

inant foot is related to factors that follow the opti-

mal strategy, and conversely, the cases with d − d

0 is related to factors that tends to depart from the

optimal strategy. We regard the cases with different

plus and minus values of d − d

due to the domi-

nant foot as having low relevance because the inﬂu-

ence of attribute is independent with respect to each

dominant foot. There are seven cases with d − d

0 for each dominant foot: position is defender (right-

footed: 0.063, left-footed: 0.011), team attribute is

club team (right-footed: 0.001, left-footed: 0.015),

ﬁrst/second kick is second kick (right-footed: 0.004,

left-footed: 0.005), order is 4 (right-footed:0.033,

left-footed: 0.076), order is 5 (right-footed: 0.007,

left-footed: 0.056), order is 6 or later (right-footed:

0.002, left-footed: 0.148), score difference is tie

(right-footed: 0.009, and left-footed: 0.010) kicker.

On the other hand, there are two cases with d − d

0 for each dominant foot: team attribute is national

team (right handed: -0.002, left handed: -0.044), and

order is 3 (right handed: -0.062, left handed: -0.100).

Do Professional Football Players Follow the Optimal Strategies in Penalty Shootout?

457

Table 5: The result of analysis.

Data attribute type

right-footed left-footed

N (p

, p

) d

d − d

N (p

, p

) d

d − d

Position

Striker 221 (0.31,0.54) 0.240 -0.078 52 (0.44,0.40) 0.230 0.071

Midﬁelder 388 (0.38,0.47) 0.144 0.058 106 (0.43,0.42) 0.230 -0.018

Defender 176 (0.42,0.44) 0.099 0.063 81 (0.41,0.40) 0.219 0.011

Nationality

Europe 365 (0.35,0.51) 0.240 -0.078 118 (0.47,0.42) 0.229 0.001

Africa 47 (0.40,0.49) 0.132 0.030 10 - - -

East Asia 167 (0.34,0.48) 0.189 -0.027 49 (0.37,0.43) 0.176 0.054

South America 147 (0.40,0.44) 0.113 0.049 34 (0.50,0.32) 0.325 -0.095

Place

Home 196 (0.33,0.51) 0.205 -0.043 60 (0.37,0.48) 0.122 0.108

Away 197 (0.37,0.47) 0.113 0.049 67 (0.48,0.37) 0.271 -0.041

Neutral 397 (0.38,0.49) 0.145 0.017 114 (0.46,0.37) 0.264 -0.034

ﬁrst/second kick

ﬁrst 405 (0.36,0.48) 0.166 -0.004 127 (0.45,0.41) 0.225 0.005

second 385 (0.37,0.49) 0.158 0.004 114 (0.45,0.41) 0.225 0.005

Score difference

Superior 173 (0.35,0.48) 0.173 -0.011 42 (0.36,0.38) 0.221 0.009

Inferior 328 (0.38,0.50) 0.164 -0.002 90 (0.42,0.41) 0.210 0.020-

Tie 331 (0.36,0.45) 0.153 0.009 109 (0.46,0.42) 0.220 0.010

Team attributes

National 229 (0.37,0.50) 0.164 -0.002 62 (0.45,0.35) 0.274 -0.044

Club 561 (0.36,0.48) 0.161 0.001 179 (0.44,0.41) 0.215 0.015

Order

1 147 (0.35,0.45) 0.161 0.001 53 (0.50,0.40) 0.272 -0.042

2 153 (0.38,0.49) 0.153 0.009 47 (0.49,0.36) 0.287 -0.057

3 157 (0.35,0.53) 0.220 -0.062 43 (0.51,0.33) 0.330 -0.100

4 143 (0.38,0.44) 0.129 0.033 47 (0.34,0.45) 0.154 0.076

5 95 (0.37,0.47) 0.155 0.007 28 (0.36,0.43) 0.174 0.056

6,7,8 95 (0.40,0.53) 0.160 0.002 23 (0.30,0.52) 0.082 0.148

Audience

50000 to 80000 250 (0.36,0.50) 0.173 -0.011 76 (0.42,0.43) 0.189 0.041

15000 to 50000 369 (0.40,0.46) 0.123 0.039 125 (0.46,0.38) 0.257 -0.027

to 15000 171 (0.31,0.53) 0.231 -0.069 40 (0.43,0.40) 0.221 0.009

GK’s height

height185cm 560 (0.37,0.49) 0.163 -0.001 156 (0.44,0.42) 0.206 0.024

height<185cm 230 (0.37,0.47) 0.158 0.004 85 (0.45,0.35) 0.273 -0.043

‘N’:The number of observation

‘-’:There are only 10 players who corresponds to this case, thus it is not speciﬁed.

5 DISCUSSION

From the pay-off matrix, it is shown that the opti-

mal strategies for right-footed and left-footed player

are to kick to right and to left respectively. From

4.5.1, right-footed player’s probability distribution

is (pR, pL) = (0.37,0.48) and left-footed player’s

probability distributionis (pR, pL) = (0.44, 0.40),

therefore it ﬁnd that the optimal strategy is dif-

ferent from the actual strategy. The left direc-

tion for the right-footed players and the right direc-

tion for the left-footed players are assumed to be

‘natural direction’. It is suggested that GK had a

high pay-off against the ‘natural direction’ kicks be-

cause GK know the actual kicker’s ‘natural direction’

in advance, and conversely, GK had a low pay-

off against the ‘unnatural direction’ kicks because

GK consider that it is rare kicker choose the

‘unnatural direction’. From the kicker’s view, kick-

ing to the ‘unnatural direction’ is the kicker’s optimal

strategy.

Why kicker’s actual action disagree with their op-

timal strategy? We hypothesize that there are two rea-

sons: one is to have a unique strategy derived from

their original theory while being aware of the optimal

strategy, and another is to choose a different strategy

considering environmental or psychological factors.

In order to identify them, we build the hypotheses

based on the d − d

in Table 5. First, we explain as-

sumptions on which the hypotheses are based, then

we build hypotheses for the factors.

5.1 The Premise on which the

Hypotheses are Based

The analytics staff is people who contribute to the

team by analyzing various data and providing the re-

sults to the players. Each professional football team

usually has analytics staff, and it is assumed that the

players obtain various information from the analytics

staff beforehand. The player who has a unique strat-

egy derived from their unique experience is a player

who ignores the analytics staff’s prior information,

thus it was excluded from this section. Therefore,

we consider the players who are willing to follow the

prior information in this section.

In teams whose precision of analytics is high and

whose precision of analytics is low, the reliability of

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

458

prior information changes, therefore the players in the

team with high precision of analytics follow the opti-

mal strategy more often. Furthermore, a player whose

skill of kicking the PK is low often follows the opti-

mal strategy based on prior information in order to

make the PK successful. Since a lot of stress also has

a large inﬂuence on the PK kicker, it becomes difﬁ-

cult to make an accurate kick, (Jordet et al., 2007),

thus the kicker who has a lot of stress tends to depart

from the optimal strategy. In summary, the premise

basing on the hypothesis identiﬁed as follows:

Prem.1 We consider the players who are willing to

follow the prior information.

Prem.2 The players in the team with high preci-

sion of analytics follow the optimal strat-

egy more often.

Prem.3 The players whose skill of kicking the PK

is low often follows the optimal strategy

Prem.4 The kicker who has a lot of stress tends to

depart from the optimal strategy.

5.2 Predicting the Cause of Following

Optimal Strategy

Here we make a hypothesis as to why players follow

the optimal strategy for each cases that have become

d − d

> 0 in Table 5. The cases are as follows

Case.1 position is defender

Case.2 team attribute is club team

Case.3 the kicking order is 4 or later.

Case.4 Score difference is tie

Case.5 ﬁrst/second kick is the second kick

Case.1 The number of players in each posi-

tion in the dataset is (StrikerMidﬁelderDefender) =

(273493257). Compared with the number of defend-

ers occupied by the formation : 4-4-2, 4-5-1, 4-3-3, 3-

5-2

which are used frequently in football, it is found

that there are few opportunities for the defender to

kick the PK

. It suggests that the skill of the defender

is low. Thus from Prem.3, the hypothesis ‘Since de-

fenders’ PK kicking skill is relatively low, they tend

to obey the optimal strategy.’ is built.

These numbers indicate the number of players in each

position : defender, midﬁelder, striker from the left

The players who participate in a penalty shootout are

only ones who are in the ﬁeld at the last moment of a match.

Case.2 There are two major differences between

club team and national team. They are the amount

of stored data and the availability of opponents’ data.

Compared with a national team, which is assembled

occasionally, a club team who has been analytics for

many years has more data. Data analysis generally

increases its precision in proportion to the amount of

data. In addition, it seems that the data used by the an-

alytics staff of a national team, especially for penalty

shootout, is mostly the data from the clubs to which

each player belong because a national team lacks data

which includes the particular situations of the penalty

shootout at a national team. Therefore, the national

team precision of analytics is lower than one of club

team. Thus from Prem.2, the hypothesis ‘Since club

teams gives the high precision of analytics, they de-

cide action according to the optimal strategy.’ is built.

Case.3 Fig.4 shows the occupancy of each position

in each kicking order. It can be seen that the ratio

of the defender in this order is increasing. Since the

defender follow the optimal strategy as described in

Case.1, the hypothesis ‘Since there are a lot of de-

fenders in the order which is 4 or later, they decide

action according to the optimal strategy.’ is built.

Case.4 The reason why the tendency of d − d

did

not appear at the time of superiority or inferiority is

that the degree of stress changes by another factor (eg,

ﬁrst/second kick). On the other hand, at the situa-

tion of the tie without temporary score difference, it

is considered that there is relatively little stress for the

kicker. Thus from Prem.4, the hypothesis ‘Since the

situation of tie gives relatively little stress, they decide

action according to the optimal strategy.’ is built.

Case.5 The Table 6 shows the value of d −d

where

the type of a kicker is classiﬁed into four categories :

ﬁrst kick& FH

, ﬁrst kick& SH

, second kick& FH,

second kick& SH. From the table, the ﬁrst kick ap-

proached the center of gravity in the SH and the sec-

ond kick approached the center of gravity in the FH,

which suggests that second kick in the FH has less

stress and the kicker second kick in the SH has a lot

of stress. As the number of people is about 1.5 times

higher in the FH, the inﬂuence of second kick in the

FH kicker comes out to the entire the kicker second

kick and it is found that it approached the center of

gravity. Thus the hypothesis ‘Since the kicker sec-

ond kick was inﬂuenced by FH the kicker second kick

FH represents kicking order is 1 to 3.

SH represents kicking order is 4 or later.

Do Professional Football Players Follow the Optimal Strategies in Penalty Shootout?

459

with less stress, they decide action according to the

optimal strategy.’ is built.

In summary, the factors that characterize kickers

who tend to follow the optimal strategy were esti-

mated as follows

Hyp.1 ‘Since defenders’ PK kicking skill is rel-

atively low, they tend to obey the optimal

strategy.’

Hyp.2 ‘Since club teams have the high precision

of analytics, they decide action according

to the optimal strategy.’

Hyp.3 ‘Since there are a lot of defenders in the

order which is 4 or later, they decide action

according to the optimal strategy.’

Hyp.4 ‘Since the situation of tie gives relatively

little stress, they decide action according

to the optimal strategy.’

Hyp.5 ‘Since the kicker second kick was inﬂu-

enced by FH the kicker second kick with

less stress, they decide action according to

the optimal strategy.’

Table 6: d − d

where the type of a kicker is classiﬁed into

four categories.

type

right-footed left-footed

N d − d

ﬁrst kick& FH 223 -0.007 77 -0.081

ﬁrst kick& SH 182 0.012 50 0.109

second kick& FH 234 0.015 66 0.010

second kick& SH 151 -0.041 48 -0.009

5.3 Predicting the Cause of not

Following Optimal Strategy

Here we make a hypothesis as to why players tends

to depart from the optimal strategy for each cases that

have become d − d

< 0 in Table 5. The cases are as

follows national team, third kicking order players.

Case.6 national team

Case.7 the kicking order is 3.

Case.6 From Section 5.2, a national team is consid-

ered to have lower precision of analytics than a club

team. Thus from Prem.2, the hypothesis ‘Since a na-

tional team has low precision of analytics, they decide

action which does not follow the optimal strategy.’ is

built.

Case.7 From the Fig.4, the ratio of midﬁelder oc-

cupies is larger than other position, which is consid-

ered to indicate that the midﬁelder has a high skill

of kicking the PK. In addition, the third kicking or-

der occupancy rate of midﬁelder is 52%, which is the

second largest proportion. The situation of defeat if

you do not score come after the third kicking order,

however, it is rare to become such situation at third

kicking order : it only happened3 times out of 157

times. Therefore, we consider that midﬁelder players

who are technically stronger but not better at handling

stress than the fourth kicking order are placed. There-

fore they feel more stress than other players, thus from

Prem.4, the hypothesis ‘Since the third kicking order

kicker has no technical problems, however weak play-

ers on the stress are placed, they decide action which

does not follow the optimal strategy.’ is built.

In summary, the factors that characterize kickers

who tends to depart from the optimal strategy were

estimated as follows

Hyp.6 ‘Since a national team has low precision

of analytics, they decide action which does

not follow the optimal strategy.’

Hyp.7 ‘Since the third kicking order kicker has no

technical problems, however weak players

on the stress are placed, they decide action

which does not follow the optimal strat-

egy.’

Figure 4: Position-Order rate.

6 RELATED WORK

6.1 Study Related to Penalty Shootout

Studies on penalty shootout have been done from var-

ious viewpoints. Geir Jordet et al (Jordet et al., 2007)

categorizes 403 penalty kicks in the three most impor-

tant international football tournaments : World Cup,

UEFA Champions League, and Copa America by the

type of event, kicking order, position, participation

time, and age and determines the success rate of each

type, they estimated factors that are inﬂuencing the

success of the PK. They concluded that psychological

effect is the most important factor.

The study also has been done to identify what kind

ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

460

of information GK uses to predict the kicker’s action.

Zhou Peiyong et al. (Peiyong and Inomata, 2012) ex-

amined the difference in predicted actions for profes-

sional football player’s PK video performed by ex-

perienced GKs and laymen. The results suggested

that experienced GKs do not rely on visual informa-

tion in front of them, therefore GK decides actions

with prior information (positions, dominant foot, etc.)

regardless of kicker’s behavior. Our study was also

done based on this assumption. However, Savels-

bergh (Savelsbergh et al., 2002) argued that visual in-

formation is also necessary. They also made a direc-

tion prediction for the experienced GKs and laymen

using the PK video kicked by the youth player of the

Netherlands league’s professional team. As a result,

it was concluded that some visual information : head,

kicking leg, non-kicking leg, ball, is the most impor-

tant element because they gazed at the head at an early

time and foot and ball at other times. However, since

this experiment only gathers gaze at these place, it

does not prove that GK is making it as the most im-

portant attributes to decide the action, thus we doubt

that these are the place that GK naturally sees in PK’s

sequence of ﬂows.

6.2 Game Theoretic Approach

Ignacio Palacios - Huerta (Palacios-Huerta, 2003)

concluded that the kicker’s optimal strategy derived

from the mixed Nash equilibrium agreed with the ac-

tual strategy taken by players by examining 1417 PK

in the professional football games. This is a differ-

ent conclusion from the one in this paper. In Ignacio

Palacios-Huerta’s study, he only deals with PKs dur-

ing the match including extra time, with the assump-

tion that each event of PK is the independent decision-

making without being affected by each other. How-

ever, it is difﬁcult for the assumption to deal with

penalty shootout because PK events are done con-

tinuously and environmental or psychological factors

such as the superiority of temporary scores must in-

ﬂuence the strategy. In his research, he also con-

sidered the strategy with only two types of direc-

tions which are summed up as ‘natural direction’ and

‘unnatural direction’, that is, i, j = C are collectively

considered by ‘unnatural direction’. On the other

hand, this paper considered three directions, there-

fore we can deal with more accurate strategy and our

study’s precision is higher.

7 CONCLUSION

In this study, we made a GK’s pay-off matrix that con-

siders the failure ratio and revealed the kicker’s opti-

mal strategy by Minimax method. Next, the presence

or absence of the gap between the actual data and the

optimal strategy was veriﬁed, and the probability dis-

tribution of each data attribute was calculated, then at-

tributes considered to be attributable to the gap were

extracted. From the attribute, we estimated the cause

of the gap between the optimal strategy and the ac-

tual strategy taken by players in the penalty shootout.

As a result of the veriﬁcation, it was suggested that

the position, team attribute, temporary scores of the

shootout and kicking order were involved in the gap.

This method can be applied to the optimal strategy in

other ﬁelds such as investment activities. In the fu-

ture, we increase the amount of the data and would

like to obtain more insight on the optimal strategy for

penalty shootout by combining the data attributes.

ACKNOWLEDGEMENTS

This work was supported by JSPS KAKENHI Grant

Numbers JP16K12411, JP17H04705.

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