Allocation of Prize Money in the Olympic-Type Tournament

Transcription

1 June 29, 2016 Allocation of Prize Money in the Olympic-Type Tournament by Pavlo Buryi Abstract We set up a game-theoretical model of an Olympic-type tournament and derive optimal amounts of effort that players choose to exert in each stage. We find that the allocation of prize money affects the distribution of effort in each stage of the tournament. When payouts increase as a player advances through the bracket, the semi-final amount of individual effort always exceeds that of the final stage. We later propose conditions on the allocation of the prize money that would enable organizers to design a desired tournament. JEL Classification: D02, C7, Z2 Key words: Nash Equilibrium, Player Performance, Sport Economics Department of Economics, The University of Tampa, pburyi@ut.edu Corresponding author: Pavlo Buryi, Department of Economics, The University of Tampa, Tampa, FL pburyi@ut.edu

2 1 Introduction Sporting events have become a center of public attention, they generate huge amounts of revenue, have significant effect on local and national economies, and sometimes even used for political purposes. Nations, states, and cities spend time, money and effort for the right to organize and host a sporting mega-event. However, sometimes the final of a much anticipated event becomes anticlimactic with a low level of effort from all participants. One of the possible explanations of such outcome is the fact that the remaining two contestants deplete the majority of their energy reserves en route to the destination final of the Australian Open, a major tennis tournament, could be used as an example. One of the finalist (Rafael Nadal) got injured midway through the contest while the other finalist Stanislas Wawrinka was simply exhausted. None of them wanted to disappoint the crowd and finished the match but it was clear that their reserves were depleted and the match-up did not meet the expectations. Another example of a similar instance is the final of the ATP Finals (the end of year tennis tournament) one of the players (Roger Federer) withdrew from the event because he was not match fit and, therefore, the most anticipated match of the tournament did not even take place. 1 There are many other examples in which the final stage of the tournament did not meet the expectations of the public, 1990 final of the FIFA World Cup between Argentina and Germany 2 is among them. It also could be the case that stages leading to the final lack effort from the participants, and therefore draw little to no attention. Regular season games of the major sport leagues such as NBA and MLB could be used as example. The question is why some professional athlete do not preserve enough energy to put up their best performance in the final stage and even more so, sometimes, not even enough to complete the match; while other treat all stages leading to the final with a lack of enthusiasm and put little to no effort at all? Could it be that the incentives were not set it in the most efficient way? Could organizers change something in order to improve the outcome of each tournament in that fashion? Moreover, inducing players to exert effort in a certain way may not be the only objective 1 The New York Times article Roger Federer Withdraws Before Year-End Final With Novak Djokovic 2 Forbes article Ranking The Last 12 World Cup Finals From The Worst To The Best labeled the final of 1990 world cup as a terrible final for a terrible tournament 1

3 of the tournament. For instance, there is unrest among players on the ATP tour regarding the payouts in the early stages of the tournament. Some players claim that since they are professionals and treat tennis as their job it is not fair that they get paid very little for their effort. A few players from the top 100 of World s ranking, state that once they subtract travel expenses, food, and coaches salaries they just break even. It could be the case that the organizers are forced to allocate a fraction of the prize money in a certain way. For example to pay travel expenses of all the players or some other kind of minimum payment. It becomes clear that extracting the maximum amount of effort is not the only objective of the organizers. Therefore it would be very helpful to know how prize money should be allocated to ensure a desirable distribution of effort, while satisfying external demands to the specifications of the tournament design. There are other examples of multi-stage contests that are frequently observed in the society. Some examples of multi-stage contests include electoral campaigns, R&D, armed conflicts, sporting tournaments, and many other. These types of tournaments have received some attention among economist 3. When organizing a multi-stage event, a tournament, or a contest one may have different objectives. Sometimes, it is desirable to induce players to exert the most effort regardless of the stage, sometimes more effort in the final stage of the tournament is desirable, but sometimes it is desirable to make each round of the tournament evenly competitive. There is a strain of literature that explores optimal allocation of prizes in a contest. It was initiated in 1902 by Francis Galton who was concerned with a way to divide 100 among a winner and a loser. Since then, there was much development in this area. There are papers that study the effect of prizes on the level of effort. For example, Moldovanu and Sela (2001) as well as Glazer and Hassin (1988) study a contest with multiple prizes where the goal of the contest is to maximize the total expected effort. Lazear and Rosen (1981) emphasize the use of relative compensation schemes in order to extract effort under moral hazard conditions. Majority of papers, including Clark and Riis (1998), Krishna and Morgan (1998), Fu and Lu (2012b), and Schweinzer and Segev (2012) among others, find that the winner-take-all principle maximizes the amount of effort. All these papers are concerned with achieving the maximum amount of effort. We are more concerned with 3 Please refer to Konrad and Kovenock (2009), Kovenock and Roberson (2012) among many others for examples on multi-battle contests. 2

4 a question of how distribution of prize money induces players to distribute effort as they advance through the bracket. To answer our research questions we set up a game-theoretical model of a contest with 4 players and find the optimal level of effort in each stage given that each player has limited reserves of energy. We also assume that all players are even or in other words are said to be identical. We find that winner takes it all type of a tournament does not extract the most effort, but instead induces players to distribute effort evenly between semi-final and final stages. To extract the most effort from all players throughout the entire tournament the payout to a runner up should be equal to the payout of a winner. 2 Model We develop a theoretical model of a tournament with 4 identical participants that compete for a championship in an Olympic type bracket. In order to win the tournament a player has to win a semi-final and then a final match-up. The bracket could be represented by the following figure. Figure 1: Olympic Type Tournament Bracket If a player loses in the first round (semi-final) he/she gets a payout of S. If a player losses in the second round (final) the payout is R, and if a player wins the tournament he/she gets W. We assume a specific function form of the probability function. The probability of winning in each stage depends on the level of effort (e) that each player exerts. For instance, the probability 3

5 that player i wins a match-up is equal to: q r i (e r i, e r o) = er i e r i +, (1) er o where q r i is a probability of player i winning his/her match-up in the round r with r = s (semi-final) or r = f (final), e r i is the level of effort that player i puts towards winning the match in the round r, while e r o is the level of effort contributed by the opponent o (o = j denotes the semi-final opponent, and o = k for the final stage opponent) in the round r. If players contribute the same amount of effort than they have 50/50 chances of winning the match-up. If one player exerts some effort, while the other one does not, then the victory of the first player is a certainty and its probability equals to 1. Each player has a strength and conditioning (energy) endowment that he/she can spend during the tournament, such that e s i + e f i = e. (2) Since we assume that players are identical, each one of them has the same endowment of energy. Regardless of how far in the tournament a player advances, he/she depletes the entire endowment of energy. Meaning, that even if the player losses in the first round, the remaining energy, which the player preserved for the second round, would be lost anyways. This happens when the cost associated with acquiring the endowment occurs prior to the tournament, which is the case with training and physical conditioning. Given payouts in each stage, probability function, and the energy constraint, each player chooses how much energy to contribute in the semi-final round. This process could be described as a two stage game. In the first stage each member of the semi-final chooses the amount of energy that he/she wants to spend in this stage. In the second stage, each winner of the semi-final match-up spends the remaining amount of energy, and the winner of the tournament is determined. When players choose how much energy to spend they are not certain of the outcome of the 4

6 match-up. Therefore, the objective of each player is to maximize the expected payout from the tournament. The expected payout is observed by each player in the first stage. With a probability (1 q s i ) player i losses in the semi-final stage and gets a payoff of S, with probability qs i player i wins his/her semi-final match-up and receives a chances to either get W or R depending on whether the player wins or losses in the second stage. The probability of success in the final stage, depends on how much energy each player has left. Regardless of how far in the tournament a player advances he/she losses all of the endowment. Expected payout to each player in the semifinal stage is equal to: E[P ayout i ] = ( 1 qi s (e s i, e s j) ) ] S + qi s (e s i, e s j) [q f i (ef i, ef k )W + (1 qf i (ef i, ef k ))R e. (3) By substituting (1) and (2) into (3) we can get es i E[P ayout i ] = S + e s i + es j ( R S + e e s i 2e e s i es k ) (W R) e. (4) We derive the optimal amount of effort that player i exerts by maximizing expected payout by setting E[P ayout i ]/ e s i = 0. The first order condition is presented below E[P ayout i ] e s i = e s j (e s i + es j )2 ( R S + e e s i 2e e s i es k ) (W R) es i e s i + es j ( (e e s k )(W R) (2e e s i es k )2 ). (5) Because we assume that players are identical we can set e s = e s i = es j = es k to derive Nash equilibrium level of individual effort in the semi-final stage of the tournament, which is equal to e s = e(w + R 2S). (6) 2(W S) By using the energy constraint (2) we can obtain the optimal level of individual effort in the final stage: e f = e(w R) 2(W S). (7) To ensure non-negative values of effort we need to set the following constraints on the parameter 5

7 values, W R, and W > S. These restrictions suggest that a player will choose a non-negative value of effort in both stages as long as a winner of the tournament is better off than a loser of the semi-final stage and no worse off than a runner up. It also can be shown that when W > R > S the amount of individual effort exerted in the semi-final match-up always exceeds that of the final. 3 Applications An organizer of a tournament may have different objectives when deciding on how prize money is distributed. We consider the following three cases. Case 1. One could be interested in making each stage of the tournament equally entertaining by making the exerted amount of individual effort equal in both stages ( e s = e f ). We can find the allocation of prize money that induces equal amount of individual effort in each stage by setting e(w R) 2(W S) e(w + R 2S) =. (8) 2(W S) It can be easily shown that e s = e f when S = R. By setting the prize money of the semi-finalist equal to that of the runner up, organizers of the tournament could achieve equal distribution of individual effort. This could be achieved by setting S = R = 0, in other words by running winner takes it all type of a tournament, or a tournament that treats all losers in the same way. When R > S, in the semi-final stage players always exert more individual effort than in the final stage. Case 2. One could be interested in extracting the maximum amount of total effort throughout the whole tournament regardless of the stage. We know that the maximum total amount of effort that could be exerted during the tournament is equal to 4e. However, in the equilibrium the total amount of effort that is spent equals to the amount of effort exerted by 4 players in the semi-final stage plus the amount of effort that 2 players exert in the final stage ( 4e s + 2e f ). The total amount of energy is equal to: e total = 4e s + 2e f = 4e(W + R 2S) 2(W S) + 2e(W R) 2(W S) 4e(3W + R 4S) =. (9) 4(W S) 6

8 Therefore, the difference between e max and e total is described by: e max e total = 4e 4e(3W + R 4S) 4(W S) = e(w R) (W S). (10) The maximum total amount of energy is spent in a tournament in which a winner gets the same prize as a runner up, W = R. In this case e s = e, meaning that each player spends all of the energy in the semi-final stage and has none left for the final. The smaller the difference in the prize money of a winner and a runner up, the more effort can be extracted from the whole tournament. A tournament in which a winner and a runner up are paid the same would extract more total effort than a winner takes it all type of a tournament. Case 3. If an organizer of the tournament wants to extract N times as much individual effort in the final as in the semi-final (Ne s = e f ), then the allocation of prize money should satisfy the following condition: S = W (N 1) + R(N + 1). (11) 2N The results are summarized by the following proposition. Proposition 1. Suppose that 4 players compete in an Olympic type tournament. Then we have the following: 1. When payouts increase as a player advances through the bracket, the amount of individual effort exerted in the semi-final stage always exceed the amount of individual effort exerted in the final. 2. When all losers receive the same prize money, the amount of individual effort exerted in semi-final stage is equal to the amount of individual effort exerted in the final. 3. When the payout to the winner is equal to the payout of the runner up, the maximum total amount of effort is extracted in the tournament, and all of energy is spent during the semifinal stage. 4. By allocating prize money in a certain way it is possible to make any stage of the tournament 7

9 more or less entertaining compared to the other stage. By using appropriate conditions along with the Prize Money constraint, an organizer of the tournament can find the way to allocate the funds. For example, if an organizer would like a final to be as entertaining as the semi-final (Case 1.) and at the same time to pay each player at least 1/X fraction of all prize money, to cover travel expenses, then the organizer of the tournament has to solve the following system of equations: P = W + R + 2S S = 1 X P S = R. Prize Money Constraint Losers get the travel expenses paid for Case 1. Even distribution of effort condition (12) By solving the system of equations we can find the following payouts that satisfy our design requirements: W = (X 3) 3 )P (13) R = S = 1 X P To ensure positive effort in both stages the following has to be true, W > S and it true if and only if X > 4. Which means that if an organizer of the tournament has to pay for travel expenses, then these expenses have to be lower than the quarter of all of the prize money. Similarly, the organizers could decide on the allocation of prize money in many other cases and see whether or not their objectives could be achieved. 4 Conclusion By allocating prize money in a certain way, it is possible to induce players to exert effort. When a winner gets no less of the prize money compared to a runner up, and the runner up - more than the semi-finalist, the amount of individual effort exerted in the semi-final would always exceed that of the final. When all losers are treated equally then the amount of individual effort exerted in the final is equal to that of the semi-final stage. Lastly, by treating the runner-up as well as the winner 8

10 of the tournament, the total maximum amount of effort could be achieved, and all of the energy would be depleted during the semi-final stage. 9

11 References Clarey, C. (2014). Roger Federer Withdraws Before Year-End Final with Novak Djokovic. The New York Times. Clark, D. J. and Riis, C. (1998). Influence and the discretionary allocation of several prizes. European Journal of Political Economy, 14: Fu, Q. and Lu, J. (2012). The optimal multi-stage contest. Economic Theory, 52(2): Glazer, A. and Hassin, R. (1988). Optimal Contests. Economic Inquiry, 26 (1): Konrad, K. and Kovenock, D. (2009). Multi-Battle Contests. Games and Economic Behavior, 66: Kovenock, D. and Roberson, B. (2012). Conflicts with Multiple Battlefields. Oxford Handbook of the Economics of Peace and Conflict (edited by Michelle R. Garfinkel and Stergios Skaperdas). Krishna, V. and Morgan, J. (1998). The winner-take-all principle in small tournaments. Advances in applied microeconomics, 7: Lazear, E. P. and Rosen, S. (1981). Rank- Order Tournaments as Optimum Labor Contracts. Journal of Political Economy, 89 (5): McMahon, B. (2014). Ranking The Last 12 World Cup Finals From The Worst To The Best. Forbes. Moldovanu, B. and Sela, A. (2001). The Optimal Allocation of Prizes in Contests. American Economic Review, 91(3): Schweinzer, P. and Segev, E. (2012). The optimal prize structure of symmetric Tullock contests. Public Choice, :