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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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