optimal feedback in contests - northwestern university€¦ · no feedback contests no-feedback...

Optimal Feedback in Contests

George Georgiadis (Northwestern Kellogg)

Jeff Ely ● Sina Moghadas Khorasani ● Luis Rayo

Ely, Georgiadis, Khorasani and Rayo Optimal Feedback in Contests Northwestern Kellogg 1 / 27

Introduction

Motivation

Contests can be an effective way to motivate effort

Workplace tournaments

Innovation races

All-pay auctions

Legal & political battles

Athletic events

We are interested in settings where the designer has an informational

advantage over participants about how they are doing mid-contest

This paper:

Characterizes optimal dynamic contest when the designer chooses when

the contest ends, how a prize is allocated, and a real-time feedback policy

Introduction

In a Nutshell: Framework

At t = 0, the principal chooses:

1 A rule specifying when the contest ends;

2 A rule specifying how the prize is allocated; and

3 A real-time feedback policy

At every t > 0, each agent chooses to work or shirk

Effort generates binary output (a Poisson success)

The principal observes successes but not efforts

Each agent observes his effort (but may learn more via feedback)

Objective: Design a contest to maximize effort given a $1 prize

or equivalently, maximize the number of successes

Introduction

Examples

Promotion tournaments

Innovation races

2006 Netflix Prize

$1M prize for developing an algorithm that predicts user film ratings

with at least 10% better accuracy than Netflix’ own algorithm

Proof-of-work / cryptocurrency mining (e.g., bitcoin)

Agents try to solve a cryptographic puzzle to win newly mined currency

By design, solution (i.e., output) follows a Poisson process

Mining new currency is inflationary ⇒ want to motivate a fixed amount

of effort with a small a prize as possible (i.e., dual problem to ours)

Introduction

In a Nutshell: Optimal Contest

Cyclical structure:

Initially, principal sets a provisional deadline T ∗

If one or more agents succeed by T ∗, the contest ends at T ∗

Otherwise, the deadline is extended to t = 2T ∗

If no agent succeeds by 2T ∗, deadline extended until 3T ∗. And so on.

When the contest ends, prize is awarded according to egalitarian rule

i.e., prize is shared equally among all agents who have succeeded

Optimal feedback policy:

Each agent is kept apprised of his own success

A deadline extension signals that no agent has succeeded thus far

Introduction

Cyclical structure:

Introduction

Cyclical structure:

Introduction

Empirical Relevance

Features of optimal design echo stylized facts from actual contests,

findings from field experiments & empirical analyses of contests

I. Cyclical structure: 2006 Netflix Prize

30-day cycles until at least one team achieved 10% target

II. Egalitarian allocation rule: Lim, Ahearne and Ham (2009)

Sales contests with either one $300 prize or five $60 prizes (↑ sales)

III. Frequent feedback about own performance: Gross (2017)

Logo-design competitions: Private feedback improves submissions

Exposing performance differences discourages continued participation

IV. Withhold feedback about rivals: Fershtman and Gneezy (2011)

60-meter footrace: Laggards would quit when running side-by-side

Introduction

Related Literature

Static tournaments / contests:

Lazear & Rosen (’81), Green & Stokey (’83), Nalebuff & Stiglitz (’83)

Optimal prize allocation: Moldovanu & Sela (’01), Drugov & Ryvkin

(’18, ’19), Olszewski & Siegel (’20)

“Turning down the heat”: Fang, Noe & Strack (’18), Letina, Liu &

Netzer (’20)

Dynamic contests:

Taylor (’95), Benkert & Letina (’20)

Feedback in contests:

“Reveal intermediate progress?”: Yildirim (’05), Lizzeri, Meyer &

Persico (’05), Aoyagi (’10), Ederer (’10), Goltsman & Mukherjee (’19)

Contests for experimentation: Halac, Kartik & Liu (’17)

Model (1/4): Players & Timing

Players: A principal and n ≥ 2 agents

At t = 0, the principal designs a contest comprising

i. a rule specifying when the contest will end,

ii. a rule for allocating a $1 prize, and

iii. a feedback policy

At every t > 0, each agent

receives a message per the feedback policy, and

chooses to work or shirk; i.e., ai,t ∈ {0,1}

When the contest ends, prize is awarded according to allocation rule

Model (2/4): Agents’ Output & “Who observes what”

Each agent’s output takes the form of a Poisson “success”:

Conditional on not having succeeded by t, an agent succeeds during

(t, t + dt) with probability ai,tdt

Who observes what:

Principal observes successes but not efforts

Each agent observes his own effort but not successes

⋆ Wolog agents can observe their own success but not those of rivals

Model (2/4): Agents’ Output & “Who observes what”

Each agent’s output takes the form of a Poisson “success”:

Conditional on not having succeeded by t, an agent succeeds during

(t, t + dt) with probability ai,tdt

Who observes what:

Principal observes successes but not efforts

Each agent observes his own effort but not successes

⋆ Wolog agents can observe their own success but not those of rivals

Model (3/4): Principal’s Choice Variables

i. A termination rule is a stopping time w.r.t each agent’s success time

e.g., contest may end at prespecified deadline, upon first success, etc

ii. A prize allocation rule specifies each agent’s share of the prize qi as

a function of the agents’ success times

e.g., prize may be awarded to first agent to succeed, randomly, etc

iii. A feedback policy specifies the message sent to each agent at every

instant as a function of the agents’ success times and past messages

e.g., each agent may be kept apprised of whether he has succeeded

Alternatives: Random feedback, feedback about others’ successes,

feedback about feedback, etc

Model (4/4): Payoffs

Given a contest, each agent’s expected payoff is

ui ,t = maxai,t∈{0,1}

E [qi − c ∫τ

0ai ,tdt] ,

where c ∈ (1/n, 1).

Principal chooses a contest to maximize total effort

max E [n

∑i=1∫

0ai ,tdt]

s.t. {ai ,t} is an equilibrium effort profilen

∑i=1

qi ≤ 1.

subject to a resource constraint.

Model (4/4): Payoffs

Given a contest, each agent’s expected payoff is

ui ,t = maxai,t∈{0,1}

E [qi − c ∫τ

0ai ,tdt] ,

where c ∈ (1/n, 1).

Principal chooses a contest to maximize total effort

max E [n

∑i=1∫

0ai ,tdt]

s.t. {ai ,t} is an equilibrium effort profilen

∑i=1

qi ≤ 1.

subject to a resource constraint.

Remarks

i. Robustness of the main result to modeling assumptions

Wolog, each agent can observe whether he has succeeded

The principal can maximize the # of successes (instead of total effort)

Agents can be uncertain about the total # of participants

ii. Constant hazard rate of success

Success during any interval depends only on effort during that interval

Extension to the case in which hazard rate increases with past efforts

Captures the idea that problem-solving ability accumulates with effort

Remarks

i. Robustness of the main result to modeling assumptions

Wolog, each agent can observe whether he has succeeded

The principal can maximize the # of successes (instead of total effort)

Agents can be uncertain about the total # of participants

ii. Constant hazard rate of success

Success during any interval depends only on effort during that interval

Extension to the case in which hazard rate increases with past efforts

Captures the idea that problem-solving ability accumulates with effort

Benchmark: No Feedback Contests

Benchmark: No-feedback Contests

Warm up: Contests without feedback and deterministic deadline

Proposition.

Optimal no-feedback contest ends at deterministic deadline T ega, and

the prize is shared equally among all agents who succeed by deadline.

In equilibrium, all agents work continuously throughout [0,T ega]

Intuition:

Principal has a limited “stock” of incentives to offer ($1 prize)

If an IC is slack at any t, contest wastes incentives, hence is suboptimal

In the proposed design, agents’ incentive constraints bind at all times

Optimal contest is essentially unique

Others exist but they are economically equivalent to the above

Proposition.

Intuition:

Proposition.

Intuition:

Proposition.

Intuition:

Towards an optimal no-feedback contest

Wolog, we can focus on contests in which:

a. An agent wins the prize with positive probability only if he succeeds

b. Working continuously on some interval [0,Ti ] is IC for agent i

Fix a contest & eq’m, and for each agent define the reward function

Ri ,t = E [qi ∣ succeeding at t] ,

which is agent’s expected share of prize conditional on succeeding at t

The egalitarian contest with deadline T has reward functions

Regai ,t = E [ 1

1 + (#rivals who succeed by T)] ,

which are time-invariant.

Regai ,t = E [ 1

Agents’ Problem

Given reward function Ri ,t , agent i ’s expected payoff

ui ,0 = maxai,t

E∫τ

0Ri ,te

− ∫ t0 ai,sds − cai ,t dt

Lemma 1.

Working continuously throughout [0,Ti ] is IC if and only if

e−tRi ,t´¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶MB at t

≥ c®

direct MC

+∫Ti

te−s Ri ,sds

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶strategic MC

for all t.

Observation: IC binds at all times only if Ri ,t is time-invariant

Agents’ Problem

Given reward function Ri ,t , agent i ’s expected payoff

ui ,0 = maxai,t

E∫τ

0Ri ,te

− ∫ t0 ai,sds − cai ,t dt

Lemma 1.

Working continuously throughout [0,Ti ] is IC if and only if

e−tRi ,t´¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶MB at t

≥ c®

direct MC

+∫Ti

te−s Ri ,sds

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶strategic MC

for all t.

Observation: IC binds at all times only if Ri ,t is time-invariant

Egalitarian Contest

Egalitarian = prize is shared equally among those who succeed

Define T ega to be the longest an agent is willing to work for if contest

is egalitarian and he expects his rivals to work for T ega units of time.

Proposition.

Any egalitarian contest with deadline T ≥ T ega has a pure-strategy

equilibrium in which all agents work throughout [0,T ega]

In any such contest, IC binds for all agents and all t ≤ T ega

Remains to show that no other contest motivates total effort ≥ nT ega

i.e, formalize argument that if IC is ever slack, contest wastes incentives

Egalitarian Contest

Proposition.

Egalitarian Contest

Proposition.

Heuristic: Any other contest wastes incentives

Any contest with different reward functions that motivates work

until T ega must have e−tRi ,t > e−tRegai ,t for at least some interval

Egalitarian contest: IC at t ′ requires that e−t′

Regai ,t′ ≥ c + 1

Alternative contest: IC at t ′ requires that e−t′

Ri ,t′ ≥ c + 1 + 2

It must therefore be the case that e−tRi ,t > e−tRegai ,t for all t < t ′

Heuristic: Any other contest wastes incentives

Recall Ri ,t is agent i ’s share of the prize conditional on succeeding at t

So any feasible contest must satisfy the “budget constraint”

∑i∫

0e−tRi ,tdt

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Pr{prize is awarded}

≤ 1 − e−nT´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¶

Pr{at least one agent succeeds}

Observation: (BC) binds in proposed contest (by the choice of T ega)

Since e−tRi ,t ≥ e−tRegai ,t for all t and the inequality is strict in at least

some interval, this alternative contest violates (BC)

i.e., this alternative contest is “too expensive”

Optimal Contest

Contests with arbitrary Feedback Policy

Key Lemma: Sufficient condition for optimality.

A contest is guaranteed to be optimal if in equilibrium:

i. The prize is awarded with probability 1

ii. Each agent earns zero rents

The principal’s objective can be rewritten as

∑i=1∫

0ai ,tdt] = 1

c[ ∑

E [qi ]

´¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Total Surplus≤1

−∑i

´¹¹¹¹¸¹¹¹¹¹¶Rents≥0

If a contest attains those bounds, it must be optimal

Note: Optimal no-feedback contest fails both criteria

Optimal Contest

∑i=1∫

0ai ,tdt] = 1

c[ ∑

E [qi ]

´¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Total Surplus≤1

−∑i

´¹¹¹¹¸¹¹¹¹¹¶Rents≥0

Optimal Contest

∑i=1∫

0ai ,tdt] = 1

c[ ∑

E [qi ]

´¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Total Surplus≤1

−∑i

´¹¹¹¹¸¹¹¹¹¹¶Rents≥0

Optimal Contest

∑i=1∫

0ai ,tdt] = 1

c[ ∑

E [qi ]

´¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Total Surplus≤1

−∑i

´¹¹¹¹¸¹¹¹¹¹¶Rents≥0

Optimal Contest

Optimal Contest (with Feedback)

Mpronto ∶= feedback policy that keeps agents apprised of own success

For some T ∗ (which I will pin down later), define

τ∗ = inf {t ∶ t = kT ∗, k ∈ N and at least one agent has succeeded}

i.e., contest runs in cycles of length T ∗ and it is terminated at the end

of the first cycle in which one or more agents have succeeded

Proposition 3.

The contest with termination rule τ∗, egalitarian prize allocation rule,

and feedback policy Mpronto is optimal.

In equilibrium, each agent works until he succeeds and earns no rents

Optimal Contest

Proposition 3.

Optimal Contest

Proposition 3.

Optimal Contest

Proof Sketch (1/2)

Each agent’s payoff at t,

ui ,t = max E∫τ∗

t[(1 − pi ,s)Ri ,s − c]ai ,sds

where pi ,t is agent’s belief that he has succeeded.

The feedback policy Mpronto implies that pi ,t ∈ {0,1}While an agent has not succeeded, pi,t = 0.

As soon as he succeeds, pi,t = 1, and he stops working

Given Mpronto , we can rewrite

t(Ri ,s − c)ai ,sds

A contest yields no rents to the agents only if Ri,t = c for all i and t

Optimal Contest

Proof Sketch (1/2)

Optimal Contest

Proof Sketch (1/2)

Optimal Contest

Proof Sketch (2/2)

Agent’s reward conditional on succeeding at t, Ri ,t , depends on how

many rivals he expects to succeed by the end of the current cycle

This number M ∼ Binom(n − 1,1 − e−T∗

), and so

Ri,t = E [ 1

1 +M] = 1 − e−nT

n(1 − e−T∗)

Pick T ∗ such that Ri ,t = c, and hence ui ,t = 0 for all i and t

Each agent weakly prefers to work until he succeeds or the contest ends

Finally, the contest does not end until at least one agent succeeds

Therefore, this contest awards prize w.p 1 and yields 0 rents to agents

(i.e., it satisfies the criteria of the Key Lemma); hence it is optimal.

Optimal Contest

Proof Sketch (2/2)

Agent’s reward conditional on succeeding at t, Ri ,t , depends on how

many rivals he expects to succeed by the end of the current cycle

This number M ∼ Binom(n − 1,1 − e−T∗

), and so

Ri,t = E [ 1

1 +M] = 1 − e−nT

n(1 − e−T∗)

Pick T ∗ such that Ri ,t = c, and hence ui ,t = 0 for all i and t

Each agent weakly prefers to work until he succeeds or the contest ends

Finally, the contest does not end until at least one agent succeeds

Therefore, this contest awards prize w.p 1 and yields 0 rents to agents

(i.e., it satisfies the criteria of the Key Lemma); hence it is optimal.

Optimal Contest

Discussion: Ingredients of optimal contest

1 To extract all rents, the feedback policy must be Mpronto

Suppose pi,t ∈ (0,1) yet (1 − pi,t)Ri,t = c. Take an interval s.t pi,t > 0

Agent can pause effort during first half of interval so pprivatei,t < peqm

Then in the second half, (1 − pprivatei,t )Ri,t > c; i.e., agent earns rents

2 The role of the egalitarian prize allocation rule:

Given Mpronto , contest yields no rents if and only if Ri,t = c for all i , t

Symmetric & time-invariant reward functions ⇐Ô egalitarian rule

3 The role of the cyclical structure:

Optimal cycle T ∗ ensures Ri,t = E [1/(1 + (#rivals who succeed))] = c

Cycles “replenish” incentives by informing agents no rival has succeeded

Optimal Contest

Extension: Increasing Hazard Rate

Increasing Hazard Rate

So far, we have assumed a constant hazard rate of success

If an agent works for t units, he succeeds with probability F(t) = 1− e−t

Assume now F is an arbitrary distribution with increasing hazard rate:

λt ∶=f (t)

1 − F(t)is (weakly) increasing

Meant to capture knowledge accumulation or a notion of progress,

whereby effort today increases the likelihood of success in the future

Increasing Hazard Rate

So far, we have assumed a constant hazard rate of success

If an agent works for t units, he succeeds with probability F(t) = 1− e−t

Assume now F is an arbitrary distribution with increasing hazard rate:

λt ∶=f (t)

1 − F(t)is (weakly) increasing

Meant to capture knowledge accumulation or a notion of progress,

whereby effort today increases the likelihood of success in the future

Increasing Hazard Rate: Optimal Contest

Proposition 4.

Assume λt ∈ (c,nc) for all t. There is an optimal contest with:

1 Stochastic cyclical structure: Each cycle ends with rate γ(λt)

2 At the end of each cycle, if a success has occurred, contest ends and

prize is awarded according to EGA rule; otherwise, a new cycle starts

3 Feedback policy Mpronto ; i.e., agents are apprised of own success

To extract all rents, we must have λtRi ,t = c for all i , t

Since λt is increasing, incentives must be frontloaded (i.e., Ri,t ↓)

Stochastic cycles: Reward from succeeding earlier in a cycle is larger

than doing so later, because agent will share prize with fewer others.

By choosing the distribution of cycle length, can fine tune Ri,t = c/λtEly, Georgiadis, Khorasani and Rayo Optimal Feedback in Contests Northwestern Kellogg 24 / 27

Proposition 4.

Application

Proof-of-Work Protocol

Proof-of-work is at the backbone of many cryptocurrencies

To add a set of transactions to the blockchain, a “miner” must first

solve a cryptographic puzzle

Doing so requires expending computing power (i.e., electricity)

As a reward, winner earns newly mined crypto-coins

By design:

Output is binary and follows a Poisson process

Arrival rate λ is determined by network and miner’s computing power

The first miner to solve the puzzle wins; i.e., contest is first-to-succeed

The idea is to make it costly to make changes to the blockchain

Application

Proof-of-Work Protocol

Proof-of-work is at the backbone of many cryptocurrencies

To add a set of transactions to the blockchain, a “miner” must first

solve a cryptographic puzzle

Doing so requires expending computing power (i.e., electricity)

As a reward, winner earns newly mined crypto-coins

By design:

Output is binary and follows a Poisson process

Arrival rate λ is determined by network and miner’s computing power

The first miner to solve the puzzle wins; i.e., contest is first-to-succeed

The idea is to make it costly to make changes to the blockchain

Application

Bitcoin: Back-of-the-envelope calculations

Some numbers for Bitcoin (as of Feb. 11, 2021):

Prize: 6.25 Bitcoins ×$47,860 ≃ $300K

State-of-the-art miner: Antminer S19 Pro (lists for $3,769)

In expectation, this miner solves the puzzle once every 26.5 years

In Illinois, it consumes electricity at rate of approx. $2,720 / year

Rents = λ × R − c = 1

26.5 years× $300K − $2,720 ≃ $8,600 / year

Dual problem: Minimize prize that implements a fixed total effort

Optimal cyclical egalitarian design:

Implements the same computing effort with prize of only 1.5 Bitcoin

Trade-off: Extract rents vs. entry and incentives to invest in equipment

Application

Prize: 6.25 Bitcoins ×$47,860 ≃ $300K

Rents = λ × R − c = 1

26.5 years× $300K − $2,720 ≃ $8,600 / year

Application

Prize: 6.25 Bitcoins ×$47,860 ≃ $300K

Rents = λ × R − c = 1

26.5 years× $300K − $2,720 ≃ $8,600 / year

Discussion

Contest design with endogenous feedback

Cyclical structure

Egalitarian prize allocation rule

Each agent is always apprised of own success, but is informed of

rivals’ successes only periodically

Future work

Decreasing hazard rate

Heterogeneous agents

More general production functions (e.g., continuous output / effort)

Discussion

Contest design with endogenous feedback

Cyclical structure

Egalitarian prize allocation rule

Each agent is always apprised of own success, but is informed of

rivals’ successes only periodically

Future work

Decreasing hazard rate

Heterogeneous agents

More general production functions (e.g., continuous output / effort)

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