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Contests Empirical and Theoretical Frameworks October 29, 2007 We examine specific types of games Games Ordinal Games Cardinal Games (Contests) • Ranking determines prize • Often simplify to 1- allocation person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort No choice of Choice of effort No choice of / cost effort / cost / cost effort / cost Efficiency of effort War of Attrition / Timing of Exits 1 choices (Nitzan) 2 (Bulow and Klemperer) Impact of superstar 3 player types (Brown) Neil Thompson & Sharat Raghavan – Oct 29, 2007 2 Examples of the key types of ordinal games Must participate Can opt out Choice of effort No choice of Choice of effort No choice of / cost effort / cost / cost effort / cost • Sports • Door prize • Oligopoly price • Standards tournaments lotteries wars competition (basketball, •HDTV golf, etc.) • Random • Media / Public •Cell phone screenings at on President technology • General Customs Bush re: Electric reward Departure of • “We’re here schemes • “Look under Alberto until I get 5 •20% up the cap to win” Gonzales volunteers” •70% flat soft drink problems •10% down promotions • University of Chicago PhD entrance policies •Accept many, then weed out Neil Thompson & Sharat Raghavan – Oct 29, 2007 3 We examine specific types of games Games Ordinal Games Cardinal Games (Contests) • Ranking determines prize • Often simplify to 1- allocation person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort No choice of Choice of effort No choice of / cost effort / cost / cost effort / cost Efficiency of effort War of Attrition / Timing of Exits 1 choices (Nitzan) 2 (Bulow and Klemperer) Impact of superstar 3 player types (Brown) Neil Thompson & Sharat Raghavan – Oct 29, 2007 4 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 Neil Thompson & Sharat Raghavan – Oct 29, 2007 5 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 Neil Thompson & Sharat Raghavan – Oct 29, 2007 6 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 Neil Thompson & Sharat Raghavan – Oct 29, 2007 7 Reducing Rent Dissipation – Model Extensions Modification Change to Base Insight Risk Aversion Increases R in equilibrium above Risk aversion causes players to the R for risk neutral players demand more R in equilibrium, so dissipation is reduced Asymmetry Asymmetric information causes Players with lower valuations lay players to have different valuations out less expenditures, so of R dissipation is reduced Uncertain Rents Positive probability that nobody Dissipation is reduced because wins R, so expected value of R is valuation / expected value of R is less than the actual prize lower than R Source of Rent Internal sources (ie losers pay Dissipation can be reduced rent) vs. external sources of rent because lobbying efforts are changes lobbying expenditures decreased (Schmidt 1992) Nature of Competitors Groups of players competing for The free riding incentive and rent creates free riding incentives smaller contest decreases rent and decreases number of players dissipation Neil Thompson & Sharat Raghavan – Oct 29, 2007 8 Reducing Rent Dissipation – Model Extensions Modification Change to Base Insight Nature of Rent Rents that are public goods tend Public goods reduce dissipation to decrease expenditures relative among rent seekers to the value to a specific player Nature of Rent-setter Rents that are set by committees Dissipation is lowered because of create high thresholds for player higher participation thresholds participation and attempts to economize expenditures Nature of Contest Contests can be modified by Leininger,Yang, Baik, and Shogren creating dynamic games with show that collusion and subgames alternating moves lower rent dissipation Multiple Rent Contests Certain scenarios enable players Research is ongoing on how these to evaluate multiple rent objects contests affect dissipation as one prize Endogenous Rent Contest Endogenous participants, rents, Research points to a reduction of parameters and order of moves rent dissipation is many of these can affect dissipation scenarios Neil Thompson & Sharat Raghavan – Oct 29, 2007 9 We examine specific types of games Games Ordinal Games Cardinal Games (Contests) • Ranking determines prize • Often simplify to 1- allocation person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort No choice of Choice of effort No choice of / cost effort / cost / cost effort / cost Efficiency of effort War of Attrition / Timing of Exits 1 choices (Nitzan) 2 (Bulow and Klemperer) Impact of superstar 3 player types (Brown) Neil Thompson & Sharat Raghavan – Oct 29, 2007 10 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 11 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,∞) h(v) = f (v) ¡F (v) - Hazard rate: 1 • Restrict attention to perfect Bayesian equilibria Note: this changes as firms drop out • Notation: - Time until a surviving firm exits: T (v ; vL, k) - Probability of being among the ultimate N survivors: P (v ; vL, k) Neil Thompson & Sharat Raghavan – Oct 29, 2007 12 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 Z game ends is: T (v; v ; 1) = v N xh(x)dx L v L - 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 13 The main result… • The unique symmetric perfect-Bayesian equilibrium is: Z v T (v; vL ; k) = c k¡1 N xh(x)dx v L • 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 14 Two special cases • General Solution: Z v T (v; vL ; k) = c k¡1 N xh(x)dx v L • 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 15 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 16 We examine specific types of games Games Ordinal Games Cardinal Games (Contests) • Ranking determines prize • Often simplify to 1- allocation person games Indivisible prizes Divisible prizes Must participate Can opt out Choice of effort No choice of Choice of effort No choice of / cost effort / cost / cost effort / cost Efficiency of effort War of Attrition / Timing of Exits 1 choices (Nitzan) 2 (Bulow and Klemperer) Impact of superstar 3 player types (Brown) Neil Thompson & Sharat Raghavan – Oct 29, 2007 17 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 18 Model is symmetric in effort • Definitions: Prize, V µe1 e2 Probabilities of winning: µe1 +e2 µe1 +e2 ei Cost of effort, • Objective functions: ¡e ¼1 = µe1 V ¡e ¼2 = e2 V 2 Player 1: µe1 +e2 1 Player 2: µe1 +e2 ¡1 0= µe2 ¡1 0= µe2 FOC: (µe1 +e2 )2 FOC: (µe1 +e2 )2 • Implications: = e = e = e¤ µ V 1 2 (1+µ)2 de¤ = 1¡µ V <0 dµ (µ+1)3 Neil Thompson & Sharat Raghavan – Oct 29, 2007 19 Effort levels increase in prize size and decrease in the skill difference ¤ e1 = e2 = e = µ V (1+µ)2 Neil Thompson & Sharat Raghavan – Oct 29, 2007 20 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 Neil Thompson & Sharat Raghavan – Oct 29, 2007 21 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 Neil Thompson & Sharat Raghavan – Oct 29, 2007 22 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… Neil Thompson & Sharat Raghavan – Oct 29, 2007 23 “If you saw this, you might not play as hard either„” Neil Thompson & Sharat Raghavan – Oct 29, 2007 24

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