Entry and disclosure in group contests

This paper analyzes information disclosure policies in group contests with best-shot effort aggregation and private entry values, finding that while disclosure always reduces aggregate investment in individual contests, within-group disclosure unambiguously increases it in group settings, whereas full disclosure yields ambiguous effects.

The Gist

Imagine a high-stakes competition where teams are trying to win a massive prize, like a giant trophy or a huge grant. But here's the twist: the prize doesn't go to the team that tries the hardest overall; it goes to the team that produces the single best idea or the single strongest effort. This is called a "best-shot" contest. Think of it like a talent show where only the best singer in the band gets the microphone, or a research lab where only the one scientist who cracks the code gets the credit, even if the whole team worked on it.

The paper by Boosey, Brookins, and Ryvkin asks a simple but tricky question: How much information should the organizers reveal to the teams before they start competing?

To understand their findings, let's break down the players, the rules, and the three different "secret rules" the organizers could use.

The Players and the Problem

Imagine you are on a team of five people. You all want to win the prize.

• The Free-Rider Problem: Since only the best effort counts, why would you try hard if you think your teammate is a genius? You might just sit back, relax, and hope they do all the work. This is called "free-riding."
• The Entry Decision: Before the competition starts, everyone has to decide: "Is my talent good enough to join this team, or should I just stay home and do something else?"

The paper looks at three ways an organizer can handle information about who is joining and how talented they are.

The Three Scenarios

1. The "Blindfolded" Game (No Disclosure)

In this scenario, nobody knows who else is on their team, and nobody knows who is on the other teams. Everyone is guessing.

• What happens: Because you don't know if your teammate is a superstar or a slacker, you can't coordinate. You might both try to be the "best" at the same time, or you might both hide in the shadows hoping someone else steps up.
• The Result: A lot of energy is wasted. People free-ride too much, and the team doesn't produce its best possible effort.

2. The "Team Huddle" (Within-Group Disclosure)

Here, the organizer tells your team, "Here is everyone on your team and how talented they are." But, you still don't know anything about the other teams.

• The Analogy: Imagine your team huddles up. You see that Sarah is the best player. So, Sarah says, "I'll take the shot; you guys just support me." The rest of the team stops trying to be the star and stops wasting energy.
• The Result: This is the winner for the organizer. By letting the team coordinate internally, they stop free-riding. The best person on the team steps up and works harder because they know they are the one carrying the load. The paper finds that this always leads to more total effort than the blindfolded game.

3. The "Public Leaderboard" (Full Disclosure)

In this scenario, the organizer reveals everything. You know who is on your team, who is on the other teams, and exactly how talented everyone is. It's like a public leaderboard showing every player's stats.

• What happens: This changes the game completely.
  • More People Join: Because everyone can see the talent levels, more people decide to join the contest (thinking they have a better chance).
  • The "Two-Horse Race": Once the competition starts, the teams realize that only the two absolute best players from all the teams combined really matter. Everyone else drops out. The game becomes a fierce duel between just two people.
• The Result: This is a mixed bag.
  • Sometimes, it leads to less effort than the "Team Huddle" because the two remaining players might get scared off or play it safe, knowing exactly who their rival is.
  • However, if the pool of talent is very deep (meaning there are many highly skilled people), this transparency can actually lead to the most effort of all. It's like a boxing match where two heavyweights know exactly how strong the other is; they might push harder than if they were guessing.

The Big Takeaway

The paper flips a common rule of thumb on its head.

• In individual contests (one person vs. one person): Revealing information usually makes people lazy and reduces the total effort. It's better to keep things mysterious.
• In group contests: Revealing information within the group (the "Team Huddle") is almost always the best thing to do. It stops people from slacking off and makes the team work harder.

However, revealing information across all groups (the "Public Leaderboard") is a gamble. It depends on how many people are competing and how similar their skills are. If the groups are small, keeping the other teams a mystery might be better. If the groups are huge and full of talent, showing everyone's stats might spark a fierce, high-effort competition.

In short: If you are running a contest with teams, tell the teams who is on their own squad so they can coordinate and stop slacking. But be very careful about showing them the stats of the other teams, because it might either supercharge the competition or cause the top players to pull their punches.

Technical Summary

Technical Summary: Entry and Disclosure in Group Contests

Problem and Setting
This paper investigates information disclosure policies in contests where competition occurs between groups rather than individuals. The authors model a generalized all-pay auction environment characterized by three distinct features:

1. Group Structure: There are $n \geq 2$ groups, each containing $m \geq 1$ potential participants.
2. Best-Shot Aggregation: Within each group, performance is determined solely by the maximum effort (the "best-shot") exerted by any member. The group with the highest best-shot wins the contest with certainty.
3. Endogenous Entry: Participation is not fixed. In an initial stage, players decide whether to enter the contest, incurring a cost (forgoing an outside option $\omega$). Players possess private valuations for the prize, drawn independently from a known distribution.
4. Public Good Prize: The prize is a group-specific public good; every member of the winning group receives their individual valuation, regardless of whether they contributed effort. This creates incentives for free-riding within groups.

The study focuses on how the timing and scope of information disclosure regarding entrants' valuations affect equilibrium entry and aggregate investment. The authors compare three disclosure regimes:

• No Disclosure (ND): Entrants know only their own valuation; the number of entrants and others' valuations remain private.
• Within-Group Disclosure (WD): Entrants observe the valuations of all members within their own group but remain uninformed about other groups.
• Full Disclosure (FD): Entrants observe the valuations of all participants across all groups.

Methodology
The authors employ a game-theoretic approach, seeking symmetric cutoff equilibria where a player enters if and only if their valuation exceeds a threshold $v^*$.

• Individual Contests Benchmark ($m=1$): The paper first establishes a benchmark for contests among individuals. In this case, WD is equivalent to ND. The analysis shows that under full disclosure, the contest reduces to a complete-information all-pay auction among the entrants.
• Group Contests ($m \geq 2$):
  • In ND, players face uncertainty about the number of competitors and their types. Equilibrium bidding functions are derived considering the probability of winning given the "free-riding" component (winning because a teammate has a higher valuation).
  • In WD, the authors assume groups coordinate to select the highest-valuation member as the active bidder (the "leader"). This solves the intra-group coordination problem, eliminating wasted effort from multiple active bidders within the same group.
  • In FD, the contest effectively becomes a complete-information all-pay auction among the group leaders. The equilibrium involves mixed strategies where only the two leaders with the highest valuations across all groups bid actively; all others bid zero.
• Comparative Statics: The authors derive closed-form expressions for expected aggregate investment under each regime and analyze the effects of the elasticity of the valuation distribution and group size.

Key Contributions and Results

1. Individual Contests Benchmark:

   • Consistent with prior literature (e.g., Morath and Münster, 2008), the paper confirms that for individual contests, full disclosure unambiguously reduces expected aggregate investment compared to no disclosure. Disclosure leads to efficiency losses and allows lower-valuation players to bid zero with positive probability.

2. Within-Group Disclosure (WD) vs. No Disclosure (ND):

   • Result: WD unambiguously increases expected aggregate investment compared to ND.
   • Mechanism: While the entry cutoff ($v^*$) remains the same in both regimes, WD solves the coordination problem within groups. In ND, multiple members may bid, wasting effort; in WD, the group coordinates on the single highest-valuation member. This reduction in free-riding is strong enough that the aggregate investment under WD exceeds even the total investment (sum of all bids) under ND.

3. Full Disclosure (FD) vs. No Disclosure (ND):

   • Result: The effect of FD on aggregate investment is ambiguous and depends on the elasticity of the type distribution and group size.
   • Mechanism: FD has three competing effects:
     1. Increased Entry: FD lowers the entry cutoff ($v^*_F < v^*$) because marginal entrants have a positive probability of winning even if their group's leader is the second highest overall (due to the mixed-strategy equilibrium of the all-pay auction). This increases the mass of entrants.
     2. Coordination: Like WD, FD solves the intra-group coordination problem.
     3. Discouragement/Inefficiency: FD reduces the contest to an auction between only two active leaders. The second-highest leader bids zero with positive probability, and the "winner" (highest leader) shades their bid significantly.
   • Condition for Dominance: FD yields higher aggregate investment than ND if the distribution of types is sufficiently elastic (specifically, if the elasticity $\xi(t) \geq \frac{1}{m-1}$). High elasticity implies that top valuations are close to each other, reducing the bid-shading effect of the second-highest leader dropping out.

4. Within-Group Disclosure (WD) vs. Full Disclosure (FD):

   • Result: The ranking is generally ambiguous. However, under the condition of full entry (fixed number of players), WD unambiguously dominates FD.
   • Reasoning: Under full entry, the entry effect disappears. The inefficiency of the FD equilibrium (where the second-highest leader often bids zero) outweighs the benefits of full information, whereas WD maintains efficient coordination without inducing the severe bid-shading of the complete-information all-pay auction.

5. Alternative Objectives:

   • The paper also examines the principal's objective of maximizing expected total investment (sum of all individual efforts) and expected highest investment.
   • WD dominates ND for total investment.
   • For expected highest investment, FD can dominate ND if the number of groups ($n$) or group size ($m$) is sufficiently large, as bid-shading in ND becomes more severe with more players.

Significance and Claims
The paper claims to be the first to study the interplay between endogenous entry, free-riding, and information disclosure specifically in the context of group contests.

• Theoretical Insight: The authors demonstrate that the standard result from individual contests—that information disclosure reduces aggregate investment—does not necessarily hold for group contests. In group settings, the benefits of solving intra-group coordination problems (via WD) and the potential for increased entry (via FD) can outweigh the efficiency losses associated with complete information.
• Design Implications: The results suggest that contest designers should generally prefer within-group disclosure (revealing information internally but keeping it opaque between groups) to maximize aggregate investment. This structure facilitates coordination without triggering the discouragement effects of full transparency.
• Nuance of Full Disclosure: While full disclosure is often detrimental in individual contests, it can be beneficial in group contests under specific conditions (large groups, elastic type distributions). However, the authors caution that full disclosure is not universally optimal and depends heavily on the distribution of types and group size.
• Robustness: The findings regarding the superiority of within-group disclosure over no disclosure appear robust even in extensions involving heterogeneous group sizes and additive aggregation technologies (though the latter requires numerical verification due to analytical intractability).

In summary, the paper argues that the optimal disclosure policy is not a binary choice between secrecy and transparency but depends critically on the organizational structure (groups vs. individuals) and the specific information architecture (within-group vs. cross-group).