contests empirical and theoretical frameworks october 29, 2007

ContestsEmpirical and Theoretical FrameworksOctober 29, 2007

2Neil Thompson & Sharat Raghavan – Oct 29, 2007

We examine specific types of games

Games

Ordinal Games (Contests)

Cardinal Games

Indivisible prizes Divisible prizes

Must participate Can opt out

Choice of effort / cost

No choice of effort / cost



• Ranking determines prize allocation

• Often simplify to 1-person games

Efficiency of effort choices (Nitzan)

War of Attrition / Timing of Exits (Bulow and Klemperer)

Impact of superstar player types (Brown)

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Examples of the key types of ordinal games






•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



Games


Cardinal Games












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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

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Base Model: Focus on Rent Dissipation

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• 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


Base Model: Symmetry and Risk Neutrality

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• 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


Reducing Rent Dissipation – Model Extensions

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Risk Aversion Increases R in equilibrium above the R for risk neutral players

Risk aversion causes players to demand more R in equilibrium, so dissipation is reduced

Asymmetry Asymmetric information causes players to have different valuations of R

Players with lower valuations lay out less expenditures, so dissipation is reduced

Uncertain Rents Positive probability that nobody wins R, so expected value of R is less than the actual prize

Dissipation is reduced because valuation / expected value of R is lower than R

Source of Rent Internal sources (ie losers pay rent) vs. external sources of rent changes lobbying expenditures

Dissipation can be reduced because lobbying efforts are decreased (Schmidt 1992)

Nature of Competitors Groups of players competing for rent creates free riding incentives and decreases number of players

The free riding incentive and smaller contest decreases rent dissipation

Change to BaseModification Insight


Reducing Rent Dissipation – Model Extensions

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Nature of Rent Rents that are public goods tend to decrease expenditures relative to the value to a specific player

Public goods reduce dissipation among rent seekers

Nature of Rent-setter Rents that are set by committees create high thresholds for player participation

Dissipation is lowered because of higher participation thresholds and attempts to economize expenditures

Nature of Contest Contests can be modified by creating dynamic games with alternating moves

Leininger, Yang, Baik, and Shogren show that collusion and subgames lower rent dissipation

Multiple Rent Contests Certain scenarios enable players to evaluate multiple rent objects as one prize

Research is ongoing on how these contests affect dissipation

Endogenous Rent Contest Endogenous participants, rents, parameters and order of moves can affect dissipation

Research points to a reduction of rent dissipation is many of these scenarios

Change to BaseModification Insight



Games


Cardinal Games












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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


Model

h(v) = f (v)1¡ F (v)

• 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


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)

T(v;vL ;1) =

Z

v

v

L

Nxh(x)dx


• 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

The main result…

T(v;vL ;k) =

Z

v

v

L

N xh(x)dxck¡ 1


• 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”

Two special cases

T(v;vL ;k) =

Z

v

v

L

N xh(x)dxck¡ 1


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



Games


Cardinal Games












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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


• Definitions:Prize, VProbabilities of winning: Cost of effort,

• Objective functions:

Player 1: Player 2:

FOC: FOC:

• Implications:

Model is symmetric in effort

¼1 = µe1µe1+e2

V ¡ e1 ¼2 = e2µe1+e2

V ¡ e2

0= µe2(µe1+e2)2 ¡ 1 0= µe2

(µe1+e2)2 ¡ 1

ei

= e2 = e¤= µ(1+µ)2 Ve1

de¤dµ = 1¡ µ

(µ+1)3 V < 0

e2µe1+e2

µe1µe1+e2


Effort levels increase in prize size and decrease in the skill difference

= e2 = e¤= µ(1+µ)2 Ve1


Brown – Econometric Specification and Results

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• 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


• 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

Brown – Robustness and Verification

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Conclusion and Superstar Evidence

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• 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…


“If you saw this, you might not play as hard either…”

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contests empirical and theoretical frameworks october 29, 2007

Documents

choice of effort cost

effort level

rent dissipation

contested rent r

r contestable rent

rent seeking contest

neil thompson sharat

contests nitzan