Contests with Revisions - University of Technology Sydney · 2019. 7. 15. · We consider two types of winner-take all contests that differ in the “marginal effectiveness of effort”

Contests with Revisions

Emmanuel DechenauxKent State University

Shakun MagoUniversity of Richmond

Informational Leakage in Competitive Settings

Bernie Sanders campaign gained access to sensitive information about Clinton

campaign during 2016 Democratic Presidential Primary due to an error by a third party

Consultant or contractor have several clients; some of whom are competitors

Potential for consultant or contractor to leak information to client about competitor’s

R&D strategy (Baccara, 2007)

Industrial Espionage

WestJet vs. Air Canada

Unilever vs. Procter and Gamble

Consider a scenario where:

initially players make choices simultaneously

but probabilistically, one player is given the chance to fully revise her choice after

seeing his rival’s choice

How does the possibility of information leakage affect expenditure in a contest?

• Impact on ex-ante expenditure (prior to leakage)?

• Incentives to revise ex-post (after leakage)?

Does the recipient of the information benefit from the ability to revise?

• Does informational leak gives a strategic advantage to the informed player?

• Does varying the likelihood of informational leak affect the degree of strategic

advantage?

We consider two types of winner-take all contests that differ in the “marginal

effectiveness of effort” or “degrees of competition.”

All-pay auction

vs.

Lottery contest

2 x 2 design that compares different degrees of strategic advantage (high vs.

low probability of informational leakage) with different degrees of

competition (all-pay vs. lottery).

Motivation and Background Model and Predictions Design and Procedures Results Conclusion

The GameTwo players A and B compete for a single prize of V.

Two rounds:

Round 1: Choose xA1 and xB simultaneously

Round 2: Type A chooses xA2; occurs with probability α

Random draw

Outcome is

determined

using xA1 and

xB

1−α α

5 / 57

Player A

chooses

xA1

Player B

chooses

xB

For Player A: she sees xB

→ can revise: chooses xA2

which may or may not be

equal to xA1

Outcome is

determined

using xA2 and xB

In the all-pay auction, the CSF is deterministic and given by

𝑝𝑖(𝑥𝑖 , 𝑥𝑗 ) = 1 if xi > 𝑥𝑗0 𝑖𝑓𝑥𝑖 < 𝑥𝑗

In the lottery contest, the CSF is probabilistic and given by

𝑝𝑖(𝑥𝑖 , 𝑥𝑗 ) =𝑥𝑖

𝑥𝑖+𝑥𝑗

All-Pay Auction: Equilibrium

Assume both players are risk neutral

Simultaneous moves game in round 1 is an asymmetric APA where Type A’s

valuation is V and Type B’s valuation is (1 −α)V (multiple equilibria)

In round 1: Both players use non-degenerate mixed strategies randomize on

the support 0, 1 − α V

In round 2: Type A wins for sure

Equilibrium expected expenditures:

𝐸 𝑥𝐴1𝐴𝑃𝐴 =

1 − 𝛼 𝑉

2𝐸 𝑥𝐵

𝐴𝑃𝐴 =(1 − 𝛼)2𝑉

2𝐸[𝑥𝐴2

𝐴𝑃𝐴] =(1 − 𝛼)2𝑉

2

Value of Flexibility

Equilibrium Predictions

V=100 Probability of Informational Leak

𝛼 = 0.25 𝛼 = 0.75

All-pay auction

Player A - Round 1 37.5 12.5

Player B - Round 1 28.1 3.1

Player A - Round 2 28.1 3.1

Probability of Revision 1 1

Value of Flexibility 36.72 91.41

Lottery

Player A - Round 1 25 25

Player B - Round 1 25 25

Player A - Round 2 25 25

Probability of Revision 0 0

Value of Flexibility 0 0

Hypothesis 1. For both levels of α, in round 1: 𝑥𝐴1𝐴𝑃𝐴 > 𝑥𝐵

𝐴𝑃𝐴 and 𝑥𝐴1𝐿 = 𝑥𝐵

𝐿

All-pay: Type A expenditure ≥Type B expenditure

Lottery: Type A expenditure = Type B expenditure

Hypothesis 2. For both types of players, 𝑥𝐴𝑃𝐴 0.25 > 𝑥𝐿 0.25 = 𝑥𝐿 0.75 > 𝑥𝐴𝑃𝐴 0.75

All-pay: Average expenditure is higher when α is lower

Lottery: Average expenditure does not depend on α

Hypothesis 3.

Frequency of revisions in all-pay auction ≥ lottery contest

In both contest, frequency is independent of α

Hypothesis 4. Value of Flexibility:

All-pay: Type A player’s average payoff ≥ Type B player’s average payoff

Lottery: Type A player’s average payoff = Type B player’s average payoff

Hypotheses

Behavior in All-pay auction and Lottery contests

Overexpenditure relative to risk neutral equilibrium in both contests (Dechenaux et al., 2015; Sheremeta, 2013)

Large Dispersion in Lottery contests (Sheremeta, 2013; Mago et al., 2013; Konrad and Morath, 2019)

Bimodal bidding in All-pay auction (Gneezy and Smordinsky, 2000)

Leader-follower games

Lottery contest: Fonseca (2009)

All-pay auction: Liu (2018); Jian et al. (2017)

Comparison of All-pay auction vs. Lottery

Davis and Reilly (1998), Faravelli and Stanca (2014); Chowdhury et al. (2019)

Strategic asymmetry: Observability and value of commitment

Bagwell (1995); van Damme and Hurkens (1997); Huck and Normann (2000); Morgan and Várdy (2004,

2007)

Experimental Design

Number

of sessions Contest

Periods

1-20

Periods

21-40

Subjects

per session

4 All-pay auction 𝛼 = 0.25 𝛼 = 0.75 8

4 All-pay auction 𝛼 = 0.75 𝛼 = 0.25 8

4 Lottery 𝛼 = 0.25 𝛼 = 0.75 8

4 Lottery 𝛼 = 0.75 𝛼 = 0.25 8

Experiment conducted at VSEEL (Purdue University)

128 subjects participated

A typical session:

Parts 1 and 2: Risk aversion and Loss aversion

Part 3: Contest, 20 periods, either α = 0.25 or α = 0.75

Part 4: Contest, 20 periods, change α

Part 5: Simultaneous moves contest for V = 0 (“joy of winning”)

Part 6: Demographics and emotions questionnaire

Lasted ≈ 90 minutes, average earnings = $25.95

Experimental Design – Each Period

• Random assignment of player types – Type A or Type B

• Random Matching

• Quasi-Strategy Method

Round 1:

Type A chooses xA1 and Type B chooses xB

Round 2:

Type A sees xB and chooses xA2 “in case round 2 is reached”

Type B makes a guess about xA2

Round 1 or round 2 is randomly drawn (based on α)

Decision Screen in Round 1

Decision Screen for Type A in Round 2

Decision Screen for Type B in Round 2

Outcome Screen

Equilibrium Comparison

Result 1A: Expenditure levels are

higher than the equilibrium prediction.

All-Pay Auction

Equilibrium Comparison

Result 1B: Expenditure levels are

higher than the equilibrium prediction.

Lottery Contest

Treatment Effects: 𝛼

All- Pay Auction Lottery Contest

Result 2: In the all-pay auction, round 1 expenditure for both players is lower when probability of

informational leakage, 𝛼 = 0.75.In Lottery contest, round 1 expenditure by both types of players is independent of 𝛼.

Treatment Effects: Player type

All-Pay Auction Lottery Contest

Result 3: In round 1, there is no significant difference in the average expenditure of type A player

and type B players.

Distribution – Lottery Contest

Player A Player B

Contrary to the subgame perfect equilibrium in pure strategies, we find individual

expenditure ranges from 0 to 100 with large standard deviations.

𝛼 = 0.25 𝛼 = 0.75

Equilibrium Observed Equilibrium Observed

Type A 25 45.71*** (31) 25 40.76** (33)

Type B 25 44.76*** (35) 25 45.49*** (34)

Distribution – All-pay auction

Player A Player B

Mass point at zero is part of the Pareto dominant equilibrium

for type B player, but not for type A player.

Type B players: anticipation of the “all-pay loser regret”

(Hyndman et al. 2012); “calm-down effect” (Konrad and

Leininger, 2007; Jian et al., 2017)

Type A players: Over-placement (Jian et al., 2017)

𝐸𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 𝑎𝑛𝑑 5

𝛼 = 0.25 𝛼 = 0.75

Type A 20%*** 37%***

Type B 34% 44%***

Bimodal bidding (Gneezy and Smorodinsky,

2006; Ernst and Thöni (2013)

Distribution – All-pay auction

Player A Player B

It is possible to construct Pareto-dominated equilibria in which

type B player places a mass point at 100.

A combination of utility of winning and fairness concerns likely

drive some of these choices, especially when 𝛼 = 0.75

𝐸𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 95 𝑎𝑛𝑑 100

𝛼 = 0.25 𝛼 = 0.75

Type A 5% 6%

Type B 8% 19%

Likelihood of Revision

Equilibrium Prediction

All-pay auction Pr (Revise| 𝛼) = 1

Lottery Contest Pr(Revise| 𝛼) = 0

Observed Rate of Revision

(a) More frequent revisions in the all-pay auction than in the lottery contest.

(b) In the all-pay auction, the frequency of revisions does not depend on the probability of informational

leakage. In the lottery contest, increasing α has a small but positive impact on the likelihood of revisions.

𝛼 = 0.25 𝛼 = 0.75

All-pay 0.89 0.86 p-value = 0.33

Lottery 0.67 0.74 p-value = 0.03

Dependent Variable

= 1 if 𝒙𝑨𝟏 is revised and = 0

otherwise

Random Effects

(1)

Round 1 variables

(2)

Subject Covariates

(3)

Dummy for α = 0.75 0.0661** 0.0846** 0.0625*

(0.03) (0.03) (0.04)

1/Period-0.0901 -0.111 -0.134

(0.08) (0.14) (0.08)

Dummy for α = 0.75 played first -0.0563 -0.0587 -0.0819

(0.06) (0.06) (0.06)

(Absolute) Distance to Best

Response

0.0107*** 0.0109***

(0.003) (0.003)

Distance to Best Response

Squared

-0.000109*** -0.000114***

(0.00003) (0.00003)

Dummy for 𝑥𝐴1 > 𝑥𝐵-0.157*** -0.165***

(0.04) (0.04)

Risk Tolerance-0.0453***

(0.02)

Loss Tolerance0.0147

(0.01)

Expenditure for Prize of Zero0.000718

(0.001)

Female-0.00392

(0.06)

Number of Observations 1280 1248 1044

Round 2: Revisions in All-Pay Auction

Rate of revision = 0.88 (vs. predicted value of 1)

Examine the difference between Player A and Player B’s

expenditure: Only 0.01, 0 and −100 should be observed

Most of the revised expenditure is equal to the best response.

Difference in

Expenditure

α = 0.25 α = 0.75

0.01 - 5 83.1 74.8

0 0.9 1.3

-100 2.5 8.2

Total 86.5 84.3

All

observations

Revised

Expenditure

Revisions in Lottery Contest

Best Response Function: 𝑅𝐴(𝑥𝐵) = 10 𝑥𝐵 − 𝑥𝐵

Regression Equation: 𝑥𝑖,𝑡 = 𝛽0 + 𝛽1𝑥−𝑖,𝑡 + 𝛽2 𝑥−𝑖,𝑡 + 𝛽3(1/𝑡) + 𝜀𝑖,𝑡

Dependent variable: 𝒙𝑨𝟐All Observations Revised Expenditure

𝜶 = 𝟎. 𝟐𝟓 𝜶 = 𝟎. 𝟕𝟓 𝜶 = 𝟎. 𝟐𝟓 𝜶 = 𝟎. 𝟕𝟓

𝑥𝐵-0.548** -0.661*** -0.607** -0.596***

(0.21) (0.17) (0.25) (0.18)

𝑥𝐵

8.090*** 10.26*** 10.04*** 10.49***

(2.02) (1.56) (2.29) (1.46)

1/Period17.38 34.95*** 18.91 22.86*

(10.82) (8.37) (11.53) (13.21)

Constant 17.85*** 11.85*** 7.423* 8.832***

(5.02) (3.91) (3.77) (3.31)

Number of Observations 640 640 434 476

R-square 0.0959 0.203 0.192 0.242


“Outspending the rival” even slightly drives a lot of

type A players’ expenditure choices.

• 72% of the choices are above type B player’s round

1 expenditure choice.

For both levels of 𝛼, revised expenditure levels are significantly higher than the risk neutral best

response function.

All

observations

Revised

Expenditure


Is the revised expenditure levels closer to the best response compared to the initial expenditure?

57% when 𝛼 = 0.25 46% when 𝛼 = 0.75

Conditional on moving in the direction of the best response, how does type A player’s round 1

expenditure compare to their rival’s?

When 𝑥𝐴1 > 𝑥𝐵 77% of the time Player A moves in the direction of the best response.

When 𝑥𝐴1 < 𝑥𝐵 33% of the time Player A moves in the direction of the best response.

Conditional on moving away from the best response, are subjects more likely to increase their

expenditure?

90% when 𝛼 = 0.25 91% when 𝛼 = 0.75

Value of Flexibility: Is Type A better off than Type B?

We define the value of flexibility as the difference between Player A’s and Player B’s expected payoffs

Δ𝐴𝑃𝐴 =𝛼𝑉

23 − 𝛼2 Δ𝐿 = 0

• The value of flexibility is higher in the all-pay auction than in the lottery contest.

• In both contests, the value of flexibility is increasing in the probability of informational leakage.

Contest 𝜶 = 𝟎. 𝟐𝟓 𝜶 = 𝟎. 𝟕𝟓

Predicted Observed Predicted Observed

All pay auction

Type A's winning rate in round 1 0.625 0.59 0.875 0.56

Type A's winning rate in round 2 1 0.96 1 0.89

Value of flexibility 36.72 19.79 91.41 66.37

Lottery



Value of flexibility 0 7.04 0 18.24

Conclusion: Study the effect of informational leakages and revisions in contests

Initial Expenditure Levels.

𝛼 has a strong impact on expenditure in the all-pay auction, but not in the lottery contest.

𝑥𝐴𝑃𝐴 0.25 > 𝑥𝐿 0.25 = 𝑥𝐿 0.75 > 𝑥𝐴𝑃𝐴 0.75

The likelihood of revision.

In the All-pay auction, likelihood of revision does not depend 𝛼In the Lottery Contest, increasing 𝛼 has a small positive impact on the likelihood of revision

Revised Expenditure levels.

In the All-pay auction, revised expenditure levels are consistent with best response.

In the Lottery Contest, revised expenditure levels are significantly higher than best response.

The value of flexibility.

In the all-pay auction, value of flexibility is positive but less than predicted.

In the lottery contest, value of flexibility is significantly higher than predicted when 𝛼 = 0.75Value of flexibility is higher in the all-pay auction vs. lottery contest

Documents

Contests with Revisions - University of Technology Sydney · 2019. 7. 15. · We consider two types of winner-take all contests that differ in the “marginal effectiveness of effort”