The economics of penalty shoot-outs
Stefan Szymanski
Economics and sport
• Economics is a theory about how people make choices
• Can we test the theory?
• We need simple problems and good data
• Sports provides an excellent laboratory for studying
decision making
Game theory
• Two players in a game seek to win
• Success for each player depends on their own actions
and the actions of the opponent
• Choices must be made in the light of what you believe
other people will do
• All economic problems have this structure
• Extreme example – herd behaviour and stock market
crashes
The penalty problem
goal
goalkeeper
ball
player
The right footed player’s natural side
goal
goalkeeper
ball
player
The left footed player’s natural side
goal
goalkeeper
ball
player
The goalkeeper’s problem
goal
Dive right
Dive left
ball
Speed and reaction time:
Well struck ball crosses the line in
0.3 seconds
player
Reaction time for ball recognition
0.2 seconds
Cannot wait to see which way the
ball is going
Probabilities (1)
goal
(Right-sided) kicker kicks to his right (R)
Goalie dives to his right (R)
Scoring probability 93% (pRR = 0.93)
Probabilities (2)
goal
Kicker kicks to his right (R)
Goalie dives to his left (L)
Scoring probability 58% (pRL = 0.58)
Probabilities (3)
goal
Kicker kicks to his left (L)
Goalie dives to his left (L)
Scoring probability 95% (pLL = 0.95)
Probabilities (4)
goal
Kicker kicks to his left (L)
Goalie dives to his right (R)
Scoring probability 70% (pLR = 0.70)
What are the chances?
• Probability of scoring if a right sided player shoots right and the
goalkeeper dives right (i.e. the wrong way): pRR = 0.93
• Probability of scoring if a right sided player shoots right and the
goalkeeper dives left (i.e. the correct way): pRL = 0.58
• Probability of scoring if a right sided player shoots left and the
goalkeeper dives left (i.e. the wrong way): pLL = 0.95
• Probability of scoring if a right sided player shoots left and the
goalkeeper dives right (i.e. the correct way): pLR = 0.70
• These probabilities are based on Palacios-Huerta (2003)
The kicker’s problem
• Shoot left or right- what should he do?
• If the kicker knew what the goalie would do it would be
easy- do the opposite!
• But the goalie doesn’t always do the same thing- there is
some probability the goalie goes right and some
probability the goalie goes left (and in this problem the
probabilities add up to 100%)
• In fact, the goalie can influence the kicker’s decision by
his choices
The kicker’s problem (maths)
• Shoot left or right- what should he do?
• The payoff to shooting right
• VKR = pGR pRR + pGL pRL
• Note this depends on what the goalkeeper choose to do!
• The payoff to shooting left
• VKL = pGR pLR + pGL pLL
The goalkeeper’s dilemma: no “pure” strategy
•
If the goalkeeper always dives right (pGR = 100%) then
1. The payoff to shooting right VKR = pRR = 0.93
2. The payoff to shooting left VKL = pLR = 0.7
•
Therefore the kicker always shoots right!
•
Likewise if the goalkeeper always dives left, the kicker always
gets a higher payoff from shooting left
•
In either case the probability of scoring is very high (0.95 or
0.93)
A “mixed” strategy
• The goalkeeper can reduce the probability of scoring by sometimes diving right
and sometimes diving left- a mixed strategy
• For example, if the goalkeeper dives right only 90% of the time, and dives left
10% of the time, then the kicker’s payoff to shooting right is now
• VKR = pGR pRR + pGL pRL = 0.9 x 0.93 + 0.1 x 0.58 = 0.9
• (i.e. less than the 93% probability when pGR = 100%)
• Note that the probability of scoring if the kicker shoots left has now increased
(from 70%) to
• VKL = pGR pLR + pGL pLL = 0.9 x 0.7 + 0.1 x 0.95 = 0.73
• By choosing a mixed strategy the goalkeeper has reduced the probability that
the kicker scores, but the kicker still has a clear preference (shoot right)
The kicker’s payoff conditional on the goalkeeper’s strategy
Probability the GK
dives right (GKR)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Payoff if the kicker
shoots right (VKR)
Payoff if the kicker
shoots left (VKL)
Kicker's optimal
strategy
0.93
0.895
0.86
0.825
0.79
0.755
0.72
0.685
0.65
0.615
0.58
0.7
0.725
0.75
0.775
0.8
0.825
0.85
0.875
0.9
0.925
0.95
R
R
R
R
L
L
L
L
L
L
L
The optimal mixed strategy
• The goalkeeper needs to choose pGR (= 1 – pGL) so that
there is no advantage to the kicker from choosing either left
or right
• This will also generate the lowest probability of scoring
• Mathematically we need VKR = VKL
• So pGR pRR + pGL pRL = pGR pLR + pGL pLL
G
• Which reduces to pR 
pLL  pRL
pRR  pRL  pLR  pLL
• In our example this would require pGR = 0.62
The kicker faces a similar kind of problem
• Kicking either always left or always right must be an inferior strategy compared
to choosing a probability of kicking left and kicking right to maximise the chance
of scoring (or minimising the chance of a save)
• Goalkeeper payoffs:
• VGR = pKR pRR + pKL pLR ;
VGL = pKR pRL + pKL pLL
• Kicker chooses pKR = 1- pKL so that VGR = VGL (goalkeeper is indifferent between
diving left or right)
• This requires pKR pRR + pKL pLR = pKR pRL + pKL pLL which implies
pRK 
pLL  pLR
pRR  pRL  pLR  pLL
• In our example this equals 0.42 (Note the similarity of the equation to pGR)
From this we derive the expected probabilities of scoring
Case 1: kicker goes right (pKR = 0.42), goalkeeper goes
right (pGR = 0.62), probability of scoring pRR = 0.93
Case 2: kicker goes right (pKR = 0.42), goalkeeper goes left
(pGL = 0.38), probability of scoring pRL = 0.58
Case 3: kicker goes left (pKL = 0.58), goalkeeper goes right
(pGR = 0.62), probability of scoring pLR = 0.70
Case 4: kicker goes left (pKL = 0.58), goalkeeper goes left
(pGL = 0.38), probability of scoring pLL = 0.95
Scoring probability when each player use the optimal mixed
strategy
• Overall scoring probability =
• pKR x pGR x pRR + pKR x pGR x pRL+ pKLx pGR x pLR + pKL x pGL x
pLL = 0.80
• The proportions for left sided kickers will be different- both
kickers and goalkeepers will choose different mixed strategies
• Note that if we believe our underlying (unconditional)
probabilities are correct we can test the theory using data
• Actual proportions should be insignificantly different from
theoretical proportions
Empirical research on penalty taking
• Palacios-Huerta, Ignacio. 2003. “Professionals Play
Minimax.” Review of Economic Studies 70, no. 2 (2003):
395–415
• Chiappori, Pierre-Andre, Timothy Groseclose and Steven
Levitt, "Testing Mixed-Strategy Equilibria When Players
Are Heterogeneous: The Case of Penalty Kicks in
Soccer." American Economic Review, 2002, 92, pp.
1138–1151
Findings
(a) Actual proportions closely match theoretical proportions for
individual kickers
(b) A vital condition for the mixed strategy to be optimal is that the
sequence chosen over time (e.g. LRLLRRRLRR…) should be
truly random
Observation from experimental studies of mixed strategiespeople are very poor at picking a random sequence (note:
LRLRLR is not random!)
But these studies found players were picking random
sequences; conclusion: when it matters people can
randomise
Ignacio Palacios-Huerta
• Professor of Economics at LSE
• Very keen football fan
• A friend of Ignacio knows Avram Grant, and through him
Ignacio sent some written advice to Grant before the
Champions League final in case a penalty shoot out took
place
Ignacio’s advice
• Ronaldo sometimes stops when approaching the kick;
when he does this he tends to shoot left most of the time
• Van der Sar (Manchester United’s goalkeeper) tends to
dive more often to kicker’s natural side than his optimal
mixed strategy would suggest and therefore Chelsea
kickers should kick more often to their unnatural side (i.e.
kick right if right-footed, kick left if left-footed)
Did Chelsea follow Ignacio’s advice?
foot
choice
Natural?
Goalie
correct?
Outcome
Score
after kick
Ballack
R
R
No
L
Yes
scores
1-1
Belletti
R
R
No
R
No
scores
2-2
Lampard
R
R
No
L
Yes
scores
2-3
Cole
L
R
Yes
L
Yes
scores
3-4
Terry
R
R
No
R
No
miss
4-4
Kalou
R
R
No
R
No
scores
5-5
kicker
choice
Natural?
Goalie
Ballack
R
R
No
L
Yes
scores
1-1
Belletti
R
R
No
R
No
scores
2-2
Lampard
R
R
No
L
Yes
scores
2-3
Cole
L
R
Yes
L
Yes
scores
3-4
Terry
R
R
No
R
No
miss
4-4
Kalou
R
R
No
R
No
scores
5-5
Anelka
R
L
Yes
R
Yes
save
6-5
kicker
correct? Outcome
Score
after kick
foot