Contests

1. Contests Empirical and Theoretical Frameworks October 29, 2007

2. We examine specific types of games Games Ordinal Games (Contests) Cardinal Games • Ranking determines prize allocation • Often simplify to 1-person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort / cost No choice of effort / cost Choice of effort / cost No choice of effort / cost Efficiency of effort choices (Nitzan) War of Attrition / Timing of Exits (Bulow and Klemperer) 2 1 Impact of superstar player types (Brown) 3 Neil Thompson & Sharat Raghavan – Oct 29, 2007

3. Examples of the key types of ordinal games Must participate Can opt out Choice of effort / cost No choice of effort / cost Choice of effort / cost No choice of effort / cost • Sports tournaments (basketball, golf, etc.) • General Electric reward schemes • 20% up • 70% flat • 10% down • University of Chicago PhD entrance policies • Accept many, then weed out • Door prize lotteries • Random screenings at Customs • “Look under the cap to win” soft drink promotions • Oligopoly price wars • Media / Public on President Bush re: Departure of Alberto Gonzales • Standards competition • HDTV • Cell phone technology • “We’re here until I get 5 volunteers” problems Neil Thompson & Sharat Raghavan – Oct 29, 2007

4. We examine specific types of games Games Ordinal Games (Contests) Cardinal Games • Ranking determines prize allocation • Often simplify to 1-person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort / cost No choice of effort / cost Choice of effort / cost No choice of effort / cost Efficiency of effort choices (Nitzan) War of Attrition / Timing of Exits (Bulow and Klemperer) 2 1 Impact of superstar player types (Brown) 3 Neil Thompson & Sharat Raghavan – Oct 29, 2007

5. Nitzan’s Survey of Rent-Seeking Contests • Nitzan analyzes strategic “winner take all” contests to measure social waste in rent seeking • Basic assumptions: • (i) contest is an N-player strategic game, N≥2 • (ii) contested rent is indivisible, ie “winner takes all” • (iii) players expend effort to increase chances of winning • Several extensions are introduced to model the effects of various constraints or modifications on the base model • The practical importance of measuring social waste or “rent dissipation” is critical for policy makers, firms or individuals in a rent seeking contest 5 Neil Thompson & Sharat Raghavan – Oct 29, 2007

6. Base Model: Focus on Rent Dissipation • N agents, R contestable rent, rent seeker i, effort level xi (same units as R) • Probability of winning R: • Where: and • Vi is rent seeker i’s payoff or expected utility • Ratio D is the relationship between total rent seeking expenditure and the value of the contested rent R • This is the crux of Nitzan’s paper – analyzing the change in the ratio by modifying the base model • Nitzan assumes two equilibriums (pure and mixed strategies) • D= and D= • Contests depend on the number and characteristics of the players, their endowments and preferences 6 Neil Thompson & Sharat Raghavan – Oct 29, 2007

7. Base Model: Symmetry and Risk Neutrality • Tullock (1980) formally introduces symmetry and risk neutrality to rent seeking contests • “Seminal contribution” – r > 0 where r is the marginal rate of lobbying outlays and the assumption that identical rent seekers are risk neutral • where • Rent dissipation increasing as the number of players increase and in the parameter r • Symmetry and risk neutrality imply that the rent is fully dissipated even when number of players is small • When can we see incomplete rent dissipation? • Risk aversion, uncertainty, heterogeneity of players 7 Neil Thompson & Sharat Raghavan – Oct 29, 2007

8. Reducing Rent Dissipation – Model Extensions Change to Base Insight Modification 8 Neil Thompson & Sharat Raghavan – Oct 29, 2007

9. Reducing Rent Dissipation – Model Extensions Change to Base Insight Modification 9 Neil Thompson & Sharat Raghavan – Oct 29, 2007

10. We examine specific types of games Games Ordinal Games (Contests) Cardinal Games • Ranking determines prize allocation • Often simplify to 1-person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort / cost No choice of effort / cost Choice of effort / cost No choice of effort / cost Efficiency of effort choices (Nitzan) War of Attrition / Timing of Exits (Bulow and Klemperer) 2 1 Impact of superstar player types (Brown) 3 Neil Thompson & Sharat Raghavan – Oct 29, 2007

11. Bulow and Klemperer (1999):Contribution to the literature • In situations of N prizes, they expand from the situation of N+1 participants to the N+k generalization • Consider wars of attrition where the ‘cost’ of the war does not end when someone drops out, only when the overall war is over • In N+1 case this is trivial since they are the same • In N+k case it changes strategies Neil Thompson & Sharat Raghavan – Oct 29, 2007

12. ( ) f v ( ) h v = ( ) F 1 ¡ v Model • N+k risk neutral firms • Cost to Firms: • ‘Fighting’: 1 unit per period • ‘After exiting’: c > 0 per period • N final firms playing receive a prize with value vi • vi is private information • vi is drawn from a distribution F(v) • F(vL) = 0 ; F(vH) = 1 ; F(· ) has strictly positive finite derivative • v Є (0,∞) • Hazard rate: • Restrict attention to perfect Bayesian equilibria • Notation: • Time until a surviving firm exits: T (v ; vL, k) • Probability of being among the ultimate N survivors: P (v ; vL, k) Note: this changes as firms drop out Neil Thompson & Sharat Raghavan – Oct 29, 2007

13. v Z ( ) ( ) h d T N 1 v ; v x x x = L ; v L Building to the main result… • Lemma 1: Firms with higher vi exit later • T (v ; vL, k) is strictly increasing in v for all vL and k • P (v ; vL, k) is probability of being in N highest firms conditional on N+k-1 firms other firms have v > vL • Lemma 2: There is at most one symmetric perfect-Baysian equilibrium of the game • Waiting times are strictly determined by firm’s vi • Lemma 3: Once only N+1 firms remain, the unique time until the game ends is: • Intuitively, this comes from setting marginal cost (1 per unit of fighting time) equal to marginal benefit (Value of win * Prob someone else has vL < vi < v) Neil Thompson & Sharat Raghavan – Oct 29, 2007

14. v Z k ¡ ( ) 1 ( ) h d N c T k x x x v ; v = L ; v L The main result… • The unique symmetric perfect-Bayesian equilibrium is: • Why is this true? • The incremental cost of waiting for the next firm to leave is c multiplied by the amount of time it will take that person to leave • The benefit is the increased probability of winning a prize • Iterate this from the k=2 case to kth case Neil Thompson & Sharat Raghavan – Oct 29, 2007

15. v Z k ¡ ( ) 1 ( ) h d N c T k x x x v ; v = L ; v L Two special cases • General Solution: • If c=0, then all but N+1 firms exit immediately • Can also be derived from the RET for 2nd Price auctions • Notice: this is not strictly an equilibrium • If c=1, the solution simplifies to the N+1 solution • Firms have no benefit from leaving early • They only consider the relative tradeoff between winning the prize and how their continuing increases game length • “Strategic Independence” Neil Thompson & Sharat Raghavan – Oct 29, 2007

16. Exit timing • The expected time between exits rises as fewer firms remain in the game • Intuition (argument about the equilibrium): • Firms that remain have higher values for the prize (Lemma 1) • To make them indifferent between staying / leaving the cost of staying must also rise • Since costs are constant per unit time, the amount of time to the next exit must increase Neil Thompson & Sharat Raghavan – Oct 29, 2007

17. We examine specific types of games Games Ordinal Games (Contests) Cardinal Games • Ranking determines prize allocation • Often simplify to 1-person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort / cost No choice of effort / cost Choice of effort / cost No choice of effort / cost Efficiency of effort choices (Nitzan) War of Attrition / Timing of Exits (Bulow and Klemperer) 2 1 Impact of superstar player types (Brown) 3 Neil Thompson & Sharat Raghavan – Oct 29, 2007

18. Summary: Adverse Incentive Effects of Competing with Superstars (Jennifer Brown) • Looks at the performance of golfers competing for prizes • Divides up her sample into exempt (higher quality) and non-exempt groups • Separates out Tiger Woods “The Superstar” • Compares each group’s performance when Tiger Woods is playing versus when he isn’t Neil Thompson & Sharat Raghavan – Oct 29, 2007

19. µ e e V V 2 1 ¡ ¡ ¼ ¼ e e = = 2 1 2 1 µ µ + + e e e e 1 1 2 2 e i µ e e 1 2 µ µ + + e e e e 1 2 1 2 µ µ e e 0 0 1 1 2 2 ¡ ¡ = = ( ( ) ) 2 2 µ µ + + e e e e 1 1 2 2 d µ 1 ¡ ¤ e V 0 < = ( ) 3 d µ µ 1 + µ V ¤ e e e = = = 1 2 ( ) 2 µ 1 + Model is symmetric in effort • Definitions: Prize, V Probabilities of winning: Cost of effort, • Objective functions: Player 1: Player 2: FOC: FOC: • Implications: Neil Thompson & Sharat Raghavan – Oct 29, 2007

20. µ V ¤ e e e = = = 1 2 ( ) 2 µ 1 + Effort levels increase in prize size and decrease in the skill difference Neil Thompson & Sharat Raghavan – Oct 29, 2007

21. Brown – Econometric Specification and Results • Data set includes 363 PGA Events from 1999-2006 • Hole by hole data available from 2002-2006 (important for variance tests) • Model: • Independent variables include: Woods presence, exempt status of a player (ie a top player) and controls for course and player attributes • Expect that the final score (strokesij) of a player will be higher when Woods is playing • Results verify hypothesis that there is a superstar effect that adversely affects performance • Exempt and non-exempt players score 0.8 strokes and 0.6 strokes higher when Tiger is playing in the same tournament 21 Neil Thompson & Sharat Raghavan – Oct 29, 2007

22. Brown – Robustness and Verification • Selection Bias • Probit model used to analyze if players avoid tournaments in which Woods plays or if they don’t make the cut in those events • Results show no selection bias (e.g. exempt players are 0-2% more likely to enter a tournament with Woods playing) • Streaks and Slumps • Estimation used to measure affect of Woods’s slumps and streaks, i.e. when he is playing below or above expectations • Results illustrate that the superstar effect increases when Woods is streaking and decreases when he is slumping • Risky Strategies & Distraction • Players may play more aggressively when Woods is participating or the extra media attention surrounding Woods could adversely affect others • Round by round data reveal that players scoring variance (a measure of “riskiness” is not different when Woods plays • Woods popularity has grown, however, the coefficient of the superstar effect has not increased over time 22 Neil Thompson & Sharat Raghavan – Oct 29, 2007

23. Conclusion and Superstar Evidence • Research into contests and other rent seeking games provide a theoretical framework for analyzing many social, industrial and competitive events • Nitzen provides a broad overview of rent seeking contests, focusing on how rent dissipation changes by modifying certain assumptions • Bulow and Klemperer show how firms react in a war of attrition and provides empirical examples such as standard settings and political coalitions • Brown uses the PGA tour and Tiger Woods as a vehicle for analyzing “the nature of competitors” as it relates to a superstar • Finally, is Tiger Woods really a superstar… 23 Neil Thompson & Sharat Raghavan – Oct 29, 2007

24. “If you saw this, you might not play as hard either…” 24 Neil Thompson & Sharat Raghavan – Oct 29, 2007