Call it a case of ‘the beautiful game’ meeting ‘A Beautiful Mind’. Russell Crowe’s portrayal of a schizophrenic Nobel laureate in the Oscar-award winning movie, has more to do with football than you think.
John Nash, the character Russell Crowe played with aplomb, was the brains behind a revolutionary mathematical idea called the Nash Equilibrium – a concept steeped in Game Theory. And that’s where our journey toward understanding how a penalty kick is simply all math, begins.
A penalty in football is the epitome of unpredictability – one which affects both teams on the field. After being introduced by William McCrum in 1891, penalty kicks have been analyzed and dissected by almost anyone who has played or watched football. The sense of unpredictability also attracts bookies, and it is common to place a bet on any penalty kick almost anywhere, legal or otherwise.
Penalties have often decided World Cups, European Cups and even heated derby matches. From Roberto Baggio in 1994 to John Terry in 2008, penalties have gotten the better of the bravest minds. Should he shoot left, or right? Or should he shoot straight? On the face of it, it appears as a totally unpredictable course of action. However, as we will see, even such an unpredictable action follows the laws of Equilibrium, more specifically the Nash Equilibrium.
Penalties in football can be thought of as a zero-sum game. That means, for one to win, the other has to lose. If we think of this in terms of the Probability Theory, assigning arbitrary values like 1 for victory and -1 for defeat, we can easily see how this is a zero-sum game. Penalty taking, in essence follows a Mixed Strategy Nash Equilibrium. That means, even if a player knows in which direction the keeper will jump, he will not change the direction of his shot. Whether that action leads to a goal or a save is irrelevant to this article as here we are trying to understand the dynamics of the underlying equilibrium.
Pure Strategy is rarely used in penalties, other than the obvious exception of lower league amateur football. Pure Strategy states that the set of all possible actions that a penalty-taker will take is finite and pre-determined. On the face of it, penalty taking might be thought of as a Pure Strategy but when we delve deeper, we can see that it is in fact a Mixed Strategy. To understand what a Mixed Strategy signifies, we assign probability values to the initial set of actions determined in the Pure Strategy phase. Mixed Strategy introduces randomness thereby increasing the unpredictability of the action taken, even though the perceived action falls under the set of all possible actions.
To assign randomness with penalty taking, the player should flip an unbiased coin and depending on that output, perform the necessary action. It might seem strange at this time, but read on. Human beings have a propensity to confuse change with randomness. The sequences of actions of human beings are predictable given the fact that we have no earlier data for any situation. For example, a penalty taker taking his very first penalty might shoot on the right and possibly let’s say, he scores. Penalty
Now, based on that single data point and nothing else, he will be taking his second penalty. A thought might pop up in his mind that since he has taken a shot to the right, he should go left now, in order to confuse the keeper or in order to balance the odds of him shooting on the right once more. And this is where we confuse change with randomness. Nowadays, the choice of a penalty-taker depends on hours and hours of analysis where coaches study the opposition keeper and find his favored side to jump. Then again, if a player is to score keeping in mind that wealth of work, he will not be following Mixed Strategy but a more advanced Totally Mixed Strategy.
In a scenario of a penalty, the player gets three choices – go left(L), go right(R) and keep straight(C). For a Mixed Strategy to take place, the three scenarios should be selected with an equal probability of 0.33. In an ideal scenario, when the data regarding the keeper is not known, the penalty-taker should choose L, R or C with equal probability. But does that mean in three consecutive shots, the penalty-taker will choose L, R, and C exactly once? No.
To understand the exact nature of the choices to be taken for three consecutive penalty shots, we need to expand the polynomial (L + R + C)^n. In this case, L=R=C=0.33 and n(no of shots)=3. This in turn means that the probability to go exactly L, R or C is 1 in 27. With the increase in n, it becomes pretty much unpredictable which way an ideal penalty-taker will shoot although this is not the scenario for human penalty-takers. As can be seen in the graph, the line moves from black to red to blue to green, signifying that the probability goes on decreasing exponentially with the increase in the value of n.
An analysis is presented below by which we can exactly calculate the probability of scoring on any side.
Assuming the following –
PS = Probability of Scoring
PL/PR/PC = Probability of choosing a side
PM = Probability of Miss
As per our assumption, we get the following –
PS + PM = 1
Extending further –
PSL = P(S|L)P(L)
PSR = P(S|R)P(R)
PSC = P(S|C)P(C)
where P(S|L) denotes the probability of scoring given that left side has been chosen
Finally, to find the exact probability of scoring in any of the three directions, we arrive at the formula In a detailed study highlighted in “Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer” by P.A. Chiappori, S. Levitt and T. Groseclose, the nature of penalty kicks is broken down into statistical terms. Following the same model, we can define two separate probabilities – one when the direction of kick (player) and the direction of jump (keeper) match and one when they don’t match. Needless to say, the probability of scoring when a keeper jumps in the wrong direction is greater than the probability of scoring when the keeper jumps in the same direction, assuming the penalty-taker does not hit the bar or posts and the ball does not go out.
For the sake of generality, let us assume all players kick in one of two direction – natural side and wrong side. For a right-footed player, it would mean shooting to the right and vice-versa for left-footed players. Data collected over a huge number of matches suggests that players always tend to go to their natural side, a fact which is not lost on the keepers. However, the chances of scoring a goal are more when one goes on the wrong side, as was seen in John Terry’s case in Moscow. Although he chose the wrong side, his decision making was spot on.
However, does this signify that the decision of the penalty-taker as well as the decision of the keeper is co-dependent? No. And this is where the beauty of Nash Equilibrium comes into play. The penalty-taker knows all possible actions of the keeper – whether he jumps to the left or to the right or he stays put. And when knowing the finite set of possible outcomes for the keeper, and assuming there is no other course of action, the penalty-taker will not benefit from changing his strategy. And that ladies and gentlemen, is the perfect example of a Nash Equilibrium.