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 player ball goalkeeper goal

The right footed player’s natural side player ball goalkeeper goal

The left footed player’s natural side player ball goalkeeper goal

The goalkeeper’s problem player ball goal Dive leftDive right Speed and reaction time: Well struck ball crosses the line in 0.3 seconds 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% (p RR = 0.93)

Probabilities (2) goal Kicker kicks to his right (R) Goalie dives to his left (L) Scoring probability 58% (p RL = 0.58)

Probabilities (3) goal Kicker kicks to his left (L) Goalie dives to his left (L) Scoring probability 95% (p LL = 0.95)

Probabilities (4) goal Kicker kicks to his left (L) Goalie dives to his right (R) Scoring probability 70% (p LR = 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): p RR = 0.93 Probability of scoring if a right sided player shoots right and the goalkeeper dives left (i.e. the correct way): p RL = 0.58 Probability of scoring if a right sided player shoots left and the goalkeeper dives left (i.e. the wrong way): p LL = 0.95 Probability of scoring if a right sided player shoots left and the goalkeeper dives right (i.e. the correct way): p LR = 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 V K R = p G R p RR + p G L p RL Note this depends on what the goalkeeper choose to do! The payoff to shooting left V K L = p G R p LR + p G L p LL

The goalkeeper’s dilemma: no “pure” strategy If the goalkeeper always dives right (p G R = 100%) then 1.The payoff to shooting right V K R = p RR = The payoff to shooting left V K L = p LR = 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 V K R = p G R p RR + p G L p RL = 0.9 x x 0.58 = 0.9 (i.e. less than the 93% probability when p G R = 100%) Note that the probability of scoring if the kicker shoots left has now increased (from 70%) to V K L = p G R p LR + p G L p LL = 0.9 x 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

The optimal mixed strategy The goalkeeper needs to choose p G R (= 1 – p G L ) 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 V K R = V K L So p G R p RR + p G L p RL = p G R p LR + p G L p LL Which reduces to In our example this would require p G R = 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: V G R = p K R p RR + p K L p LR ; V G L = p K R p RL + p K L p LL Kicker chooses p K R = 1- p K L so that V G R = V G L (goalkeeper is indifferent between diving left or right) This requires p K R p RR + p K L p LR = p K R p RL + p K L p LL which implies In our example this equals 0.42 (Note the similarity of the equation to p G R )

From this we derive the expected probabilities of scoring Case 1: kicker goes right (p K R = 0.42), goalkeeper goes right (p G R = 0.62), probability of scoring p RR = 0.93 Case 2: kicker goes right (p K R = 0.42), goalkeeper goes left (p G L = 0.38), probability of scoring p RL = 0.58 Case 3: kicker goes left (p K L = 0.58), goalkeeper goes right (p G R = 0.62), probability of scoring p LR = 0.70 Case 4: kicker goes left (p K L = 0.58), goalkeeper goes left (p G L = 0.38), probability of scoring p LL = 0.95

Scoring probability when each player use the optimal mixed strategy Overall scoring probability = p K R x p G R x p RR + p K R x p G R x p RL + p K L x p G R x p LR + p K L x p G L x p LL = 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 “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 strategies- people 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? kicker footchoiceNatural?Goaliecorrect?Outcome Score after kick Ballack RRNoLYesscores1-1 BellettiRRNoR scores2-2 LampardRRNoLYesscores2-3 ColeLRYesL scores3-4 Terry RRNoR miss4-4 KalouRRNoR scores5-5

kicker footchoiceNatural?Goaliecorrect?Outcome Score after kick Ballack RRNoLYesscores1-1 BellettiRRNoR scores2-2 LampardRRNoLYesscores2-3 ColeLRYesL scores3-4 Terry RRNoR miss4-4 KalouRRNoR scores5-5 AnelkaRLYesR save6-5