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Abstract

We study the volunteer’s dilemma if the set of possible volunteers is partitioned into small teams that can coordinate internally but not across teams. If coordination is costless, the aggregate equilibrium payoff is determined by the number of teams only. The team size is relevant for individual players’ payoffs. They prefer larger teams. The optimality of the team structure is also considered if the coordination cost in teams is an increasing and strictly convex function of team size. Such costs can make very large teams suboptimal. Also, such costs suggest a partitioning into teams of roughly equal size.

Keywords

Discrete public good partial coordination partition into teams public good provision volunteer’s dilemma D72 H41

Introduction

In the center of Berlin near Potsdamer Platz, a plaque honors Nikolai I. Massalov, a sergeant in the USSR army. On April 30, 1945, during the battle for Berlin, Massalov voluntarily risked his own life to save a small child from a burning house. In doing so, he gave in to his own impulse and at the same time presumably brought relief to other bystanders who had also heard the little child screaming.¹ This and other situations are based on a common structure that is generally known as the volunteer’s dilemma. In abstract terms this problem has been studied in a static game with complete information by Diekmann (1985) and as a dynamic game with incomplete information by Bliss and Nalebuff (1984). Several people are faced with the decision problem of whether to volunteer or not, that is, whether to carry out a costly task that benefits all members of the group. Carrying out the task is the provision of a discrete public good: Each person can resolve the existing problem, benefitting the whole group by taking costly own action. The cost is lower than the benefit to the volunteer themself if the problem is resolved. Hence, each player would be willing to act if no one else does. But everyone would still rather see another volunteer do the job. So who and how many then choose to volunteer? Because the players have to decide independently and simultaneously whether to take action or not, a coordination problem arises. Often there will be no volunteers at all because everyone is relying on someone else to do the job. Or several players become active independent of each other, so the task is completed, but with a redundancy in effort.

The contributions by Diekmann (1985) and Bliss and Nalebuff (1984) on this dilemma sparked a large literature on volunteering. Prominent examples that have been discussed are helping injured persons (Bergstrom, 2017); sanctioning the violation of social norms (Nguyen, 2020); making third parties aware of grievances that can easily be addressed and, if addressed, will resolve a problem for members of an entire group (Franzen, 1995); individual members of a group taking on the risky task of warning the entire group of an emerging danger (Archetti, 2011); and volunteering by members of a lobby group in the interests of the group (Glazer and McMillan, 1990). Barbieri et al. (2020) place the lobbying example into a context of competition between several such lobby groups to preempt each other, and they provide anecdotal evidence. Konrad (2024) analyzes a military context, in which individual defenders can volunteer and choose to eliminate single enemies in a preemptive attack. García and Traulsen (2019) discuss volunteering problems in the field of biology.²

Successful provision of the discrete public good might sometimes need contributions by more than one contributor. For an early study of such problems of provision of a discrete public good see the seminal contribution by Palfrey and Rosenthal (1984). The need for more than one volunteer also attracted attention in the context of evolutionary biology (see, e.g., Nöldeke and Peña, 2020; Peña et al., 2024). Archetti (2011) and Friehe and Tabbach (2018) investigate the role of uncertainty, for example, when the action of each individual volunteer only eliminates the problem with a certain probability. The most comprehensive recent review on the provision of discrete public goods is by Barrachina and Jiménez-Fernández (2024) and also mentions the literature on the volunteer’s dilemma but leaves much of its discussion for future research.

The characteristic feature in Diekmann’s (1985) volunteer’s dilemma is the complete inability of the potential contributors to coordinate their actions. We study partial coordination as a departure from this benchmark model. Compared to an unstructured set of completely uncoordinated players, we study a set of potential contributors that has a particular internal structure. The set is partitioned into non-overlapping subsets of players, which will be called “teams”. The members of each team can use a means to communicate or use other techniques to coordinate the actions inside the team, whereas the coordination of actions across different teams is not possible. The guiding research questions are how the coordination inside teams affects volunteering in equilibrium, how beneficial this team structure is overall, and which team members gain or lose. A further section addresses the desirability of team structures if establishing the tool to coordinate inside a team has a cost that depends on the size of the team.

Team substructures among the set of players have been analyzed in games other than the volunteering game. Early contributions by Hausken (1995) and Rapoport and Amaldoss (1999) generated a large literature on conflict games between teams. In such games, the contributions of a member of one group tend to benefit the members of his or her group but harm the players in other groups. A key difference between such inter-alliance contests and the volunteering problem here is that, in the volunteer’s dilemma with teams, the costly volunteering by a member of one team does not harm but benefits all players, including the members of all other teams.

Welfare effects of coordination in subgroups of a larger set of players have been studied in areas and strategic situations other than the volunteer’s dilemma. The benefits of coordination in the group for its members and for non-members vary greatly in these contexts. The work of Salant et al. (1983) is one of the first analyses of this issue and is carried out in the institutional context of cooperation/mergers among firms that compete in the same industry or market. In Cournot markets, a cooperation/merger of a subgroup of firms is typically bad for the merging firms, but bystanders benefit. Their result is sensitive to the type of firm competition assumed. For firms with differentiated goods, and in a Bertrand competition setting, the analysis by Deneckere and Davidson (1985) shows that coordination in a subgroup is often beneficial for both subgroup members and non-members. Analogous questions arise in the context of defence alliances (Bruce, 1990; Esteban and Sákovics, 2003; Ihori, 2000), and in the context of cooperation between individual countries in global tax competition (Konrad and Schjelderup, 1999). The examples illustrate that the advantageousness of coordination/cooperation in subgroups varies depending on the specific strategic context. This ambivalence is precisely what makes it interesting to study how a partition into teams and the possibility of coordination within teams affects the outcome in Diekmann’s (1985) volunteering game.

The feasibility of coordination inside subgroups may have many reasons. In the example of the child rescue by Sergeant Massalov described above, one might imagine that the soldiers are organized into small teams that fight in immediate proximity and can talk directly to each other. However, communication across such teams to rescue a child is logistically difficult under conditions of military street combat and prevents coordination. Many other determinants of such team boundaries and resulting coordination obstacles are conceivable and realistic. In a set of players who speak several languages the teams with internal coordination might easily emerge along the lines of players who speak the same language. Other characteristics that define the boundaries of teams can be religious or cultural commonalities. Players could also be partitioned into different institutions with clear boundaries for other, exogenous reasons. Suppose, for instance, that groups of players work in different firms. The costs of communication-based coordination between the players might be low within one organizational unit or firm, but high across units or firms. In all such cases, coordination within teams might therefore be feasible or easy, but coordination between teams might be prohibitive.

The subsequent section analyzes the impact of an existing team structure for the volunteering outcome and consists of five subsections. The first subsection describes the game-theory framework and its key elements. The second subsection derives the Nash equilibrium, assuming ex-ante symmetry among players inside each team. It also shows how the framework nests the original analysis by Diekmann (1985) and answers whether the coordination inside the teams makes it more or less likely that a player volunteers. The third subsection derives the welfare properties of the volunteering equilibrium and how welfare depends on the overall number of players and on the size of the teams. The fourth subsection studies the role of asymmetry in the size of the teams. It also considers the benefits and costs of a player if the player could choose to leave one team in a given partition and stand alone. Additionally, it compares the outcomes for two arbitrary partitions of a given set. The results obtained implicitly answer questions about the stability of partitions, showing that team members have no incentives to leave a given team and stand alone. Furthermore, partitions with fewer but larger teams tend to be beneficial and advantageous in the absence of considerations about coordination costs. The structure of the partition into teams might be endogenous and could be chosen with the aim to maximize welfare. The last subsection studies the properties of an optimal partition into teams, accounting for the cost of establishing a coordination tool inside each team. The final section concludes and summarizes the research results.

The game-theory model

The formal framework

We revisit and extend Diekmann’s (1985) volunteering problem in a game-theoretic model with complete information and common knowledge. Let m + 1 potential volunteers form the group M. Much like in the examples in the introduction, the set M is partitioned into disjoined subsets (“teams”) denoted by M₁, …, M_K. This partitioning, and partial coordination inside each team is the main departure from Diekmann (1985). Team M_k has m_k members, a representative element of team M_k is denoted by j_k , and m₁ + … m_K = m + 1, with m₁ ≥…≥ m_K ≥ 1 by how the teams are numbered. The partition into teams is exogenously given, by technological, institutional, or geographical constraints, as discussed in the context of examples in the Introduction. The meaning of this partition is that the members of a team can coordinate their volunteering behavior, whereas players cannot coordinate with players across the boundaries of their team.

Each player j_k ∈ M_k can choose only between two possible actions. Player j_k’s action is denoted by $θ_{j_{k}} \in {0,1}$ . Here, $θ_{j_{k}} = 0$ denotes the action to not volunteer, and $θ_{j_{k}} = 1$ the action to volunteer. Each player who volunteers has a cost of c. If at least one player volunteers, a discrete public good is provided. This outcome gives all m + 1 players the same benefit of size b. All players choose their action simultaneously and independently.

In each team M_k a coordination mechanism³ operates as follows: There is a random device described as an urn that contains m_k − 1 balls with a “0” on it, and one ball with a “1” on it. The random mechanism assigns one of these balls to each team member j_k ∈ {1, …, m_k}, which might be thought of as drawing balls until the urn is exhausted. Members inside the team learn which number is on the ball they draw. We denote the number on the ball of player j_k by $μ_{j_{k}}$ , with either $μ_{j_{k}} =$ “0” or $μ_{j_{k}} =$ “1”. We further assume that the draws taking place in different teams are stochastically independent events. The numbers on the balls do not force the players to take particular actions. But they can be used by the team members to form beliefs about the choices of the co-players in their team. We write the (possibly mixed) strategies of the members j_k of team k as functions $s_{i_{k}}$ : $μ_{j_{k}} \to p_{j_{k}}$ , that map the number $μ_{j_{k}}$ on j_k ’s ball into the probability $p_{j_{k}} \in [0,1]$ that j_k volunteers.

Finally, let us consider payoffs. A benefit of size b is obtained by each player if and only if at least one of the players from M chooses $θ_{j_{k}} = 1$ . In terms of public good theory, volunteering provides a discrete public good. It is provided if at least one player chooses $θ_{j_{k}} = 1$ and otherwise not. Recall that players who volunteer bear a cost of volunteering equal to c, and non-volunteers bear no cost.⁴ For the analysis of the volunteering equilibrium we take the team structure as given. We assume that 0 < c < b = 1. The first inequality c > 0 is important, as for c = 0 all players could simply volunteer at no cost. The second inequality makes sure that each player is willing to provide the public good if no other player does that. And for simplicity we normalize b = 1. Homogeneity of the cost parameter and the benefit parameter across players is for better comparability with Diekmann’s (1985) analysis. All this gives a player a final payoff of 0 if no volunteer appears, of 1 − c if the player volunteers, and of 1 if this player did not volunteer but one or several of the other players do volunteer⁵. And all this is common knowledge.

We study the volunteering equilibrium, its welfare properties, and its comparative static properties for exogenously given teams which solved the coordination problem in this section. If there are costs to establish teams that can coordinate inside, these coordination costs are given and sunk at this point and can be ignored in this section. We turn to issues of endogenous team structures, focusing on the role of the cost of enabling coordination inside teams in the subsection directly preceding the Conclusions section.

Equilibrium

We first study the reference case, that is, the case originally analyzed by Diekmann (1985), which is nested in the framework here. Our framework decreases to this benchmark for m₁ = m₂ = … = m_K = 1 and K = m + 1. For the benchmark it has been discussed that, in the absence of a coordination device, it is most reasonable to focus on the symmetric equilibrium among the m + 1 players:

Proposition 1

(Diekmann, 1985) The symmetric equilibrium of a set of m + 1 single players is in mixed strategies. Each player volunteers with a probability $p = 1 - \sqrt[m]{c}$ . The overall probability for provision is P = 1 − c^(m ⁺ ¹⁾^/m. Both p and P are decreasing in m and P reaches 1 − c in the limit if the set of possible contributors becomes very large.

Proof

Let p be the uniform equilibrium probability of volunteering for all players other than j_k. Then player j_k is indifferent as to whether to volunteer or not if

(1 - c) = 1 - {(1 - p)}^{m} .

(1)

Solving for p yields

p = 1 - \sqrt[m]{c} .

(2)

This shows that volunteering and not volunteering and any mixing between these two choices are among j_k’s optimal replies, and this set of optimal replies includes $p_{j_{k}} = p$ as in (2). It follows that the probability that no player will volunteer is 1 − P = (1 − p)^m+1. Solving for P and using (2) yields

P = 1 - c^{\frac{m + 1}{m}} .

(3)

Calculus shows that both p and P decrease in m, with P reaching a limit of

\lim_{m \to \infty} P = \lim_{m \to \infty} (1 - c^{\frac{m + 1}{m}}) = 1 - c .

(4)

The focus on the symmetric mixed-strategies equilibrium in Proposition 1 is appealing due to the property that all players are identical along all relevant dimensions, and due to the absence of any means to coordinate across players. Of course, there are very many asymmetric equilibria even in the Diekmann (1985) model. The set of equilibria includes, for instance, equilibria in which one of the m + 1 players volunteers with probability 1, and all others free-ride. There are also equilibria in which several players never volunteer, and in which the remaining players choose a symmetric mixed strategy.

This multiplicity clarifies that there exist multiple equilibria in the game with a partition into multi-player teams. We focus on equilibria that emerge if we assume that players who are in an equivalent situation behave symmetrically, that there is no means to coordinate across the boundaries of teams, and that players inside each team use a coordination device that allows them to avoid duplicating the volunteering effort inside the team. This leads to the following proposition:

Proposition 2

A Nash equilibrium exists in which players j_k volunteer with a probability of

p_{j_{k}} (μ_{j_{k}}) = \{\begin{cases} 1 - \sqrt[K - 1]{c} & i f μ_{j_{k}} = 1 \\ 0 & i f μ_{j_{k}} = 0 . \end{cases}

(5)

It holds that the probability of at least one player volunteering in this equilibrium is

P = 1 - c^{\frac{K}{K - 1}} .

(6)

Proof

In the candidate equilibrium one single player (the one with $μ_{j_{k}} = 1$ ) in each team volunteers with probability $1 - \sqrt[K - 1]{c}$ , whereas all other players in this team do not volunteer. To see that these strategies are mutually optimal replies, consider player j_k in team k . Suppose the player has $μ_{j_{k}} = 1$ . The player rightly anticipates that none of the other members in team k will volunteer and anticipates the probability $p_{k^{'}} = 1 - \sqrt[K - 1]{c}$ under which one player j_k_′ in each other team k′ will volunteer. Player j_k’s payoff from own volunteering is (1 − c). The payoff from not volunteering is equal to the probability that at least one other team volunteers. Hence, j_k’s payoff from not volunteering is $1 - {(1 - (1 - \sqrt[K - 1]{c}))}^{K - 1}$ . The player is just indifferent if

(1 - c) = 1 - {(1 - (1 - \sqrt[K - 1]{c}))}^{K - 1} .

(7)

This makes mixing with probability $p_{k} = 1 - \sqrt[K - 1]{c}$ one of this player’s optimal choices.

Consider next another player $j_{k}^{'}$ inside team k. This player drew $μ_{j_{k}^{'}} = 0$ and, from candidate equilibrium play, anticipates that there is one other player inside team k and one player in each other team who will volunteer with probability $1 - \sqrt[K - 1]{c}$ , whereas all other players choose not to volunteer. If player $j_{k}^{'}$ volunteers, the player has a payoff equal to 1 − c. If this player does not volunteer, anticipating the equilibrium behavior of all other players, then $j_{k}^{'}$ ’s expected payoff is

1 - {(1 - (1 - \sqrt[K - 1]{c}))}^{K} = 1 - c^{\frac{K}{K - 1}} > 1 - c .

(8)

So it is strictly better for a player with $μ_{j_{k}} = 0$ to not volunteer. Analogous arguments apply to the choices in all teams. So, the candidate equilibrium strategies are mutually optimal replies.

We can use the candidate equilibrium strategies to calculate the probability of a player volunteering and the overall probability of at least one player volunteering. Before a player j_k knows $μ_{j_{k}}$ , the probability of j_k volunteering is $(1 / m_{k}) (1 - \sqrt[K - 1]{c})$ . After the own $μ_{j_{k}}$ is observed: The volunteering probability for a player who has $μ_{j_{k}} = 0$ is zero, the probability for a player who has $μ_{j_{k}} = 1$ is $1 - \sqrt[K - 1]{c}$ . As choices of players of different teams are stochastically independent, the overall probability of at least one player from the set of all players volunteering is

P = 1 - c^{\frac{K}{K - 1}} .

(9)

The equilibrium in Proposition 2 is not unique, for reasons similar to the ones discussed in the context of the Diekmann (1985) case. For instance, all players might simply ignore their $μ_{i_{k}}$ and play the fully non-coordinated equilibrium. The coordination mechanism plays an important role in achieving the equilibrium in Proposition 2. Making their strategies non-trivial functions of their $μ_{j_{k}}$ allows them to pick one team member as the one who might volunteer, and avoids duplication of volunteering effort inside the team.

The equilibrium has interesting and attractive properties.

(1) For the given anticipated equilibrium strategies of players in other teams, the strategies taken by the team obey symmetry: Players within the team who receive the same $μ_{j_{k}}$ use identical strategies.

(2) Due to within-team coordination, all but one player in this team do not volunteer. Each of them is better off than in the Diekmann equilibrium in Proposition 1, whereas the player who volunteers with positive probability has the same payoff as in the Diekmann-equilibrium in Proposition 1.

(3) Coordination as in Proposition 2 causes each team to behave like one single player in a volunteer’s dilemma with a set of K players. Inside the team, the single player who might volunteer is randomly decided as part of the coordination mechanism. Once this is established, the solution in Proposition 2 is analogous to that in Diekmann (1985) for the set of players that volunteer with a positive probability.

(4) Probability P depends only on the number of teams. It does not depend on the specific partition of the set of players into coordination teams, the specific size that these teams have, or asymmetries in the size of the different teams.

(5) The probability that at least one player volunteers, that is, that the public good will be provided, is decreasing in the number of teams.

Welfare

The starting point of a welfare analysis is again the Diekmann model. An efficient solution would be reached there if only one of the m + 1 players volunteers, leading to an aggregate welfare of (m + 1) − c. The equilibrium properties suggest that each player has the same expected welfare of (1 − c) , such that overall welfare is (m + 1) (1 − c). This is the same welfare that emerges if all players choose to volunteer and the welfare loss is as big as the redundant volunteering costs by m players. Hence, the expected welfare loss in the equilibrium in Proposition 1 is mc.

The situation with teams is more complex. The normative benchmark is again maximum welfare: (m + 1) − c. The sum of the expected payoffs in the equilibrium in Proposition 2 can be calculated in two steps. Players j_k with $μ_{j_{k}} = 1$ are indifferent between volunteering or not, such that for K players the expected payoff is determined by (7) and equal to (1 − c). For players j_k with $μ_{j_{k}} = 0$ , by (8) not to volunteer is an equilibrium reply that is strictly superior to volunteering: They all have a payoff larger than (1 − c). They do not volunteer and receive the public good, and the benefit from it, with a probability of P = 1 − c^K/(K−1). So their payoff is higher by

(1 - c^{\frac{K}{K - 1}}) - (1 - c) = c - c^{\frac{K}{K - 1}} .

(10)

The welfare gain from the coordination in teams for strict non-volunteers (i.e., players j_k with $μ_{j_{k}} = 0$ ) is the same for each of them, regardless of whether the player belongs to a small or a large team. For a given number of teams, there are more strict non-volunteers in larger teams than in smaller teams. Before $(μ_{1_{k}}, \dots μ_{{(m_{k})}_{k}})$ is drawn for team k, the likelihood of becoming a strict non-volunteer is larger for a single member if k is a larger team. The number of teams determines the welfare gain compared to the Diekmann’s (1985) equilibrium, but symmetry or asymmetry in the size composition of teams does not affect overall welfare. As Proposition 2 shows, each team has one member who might volunteer and this one player does this with the same probability in all teams. That implies that the overall probability of the provision of the public good depends on the number K of teams, but is independent of how the players are allocated between the K teams.

Still, differences in the size of the teams have implications for the ex ante expected payoffs of their members. Consider two teams M_k and M_h. Let m_h ≤ m_k. Members of team k and team h volunteer, on average, with probabilities p/m_k and p/m_h with p/m_k ≤ p/m_h. The expected contribution burden for a team member is smaller if the team has more members.

Consider taking one member j_h from the smaller team h and putting j_h inside the larger team k. The relocation has implications not only for the relocated member, but also for the members who are not relocated. Players who stay in team h will have a higher expected volunteering probability, players who stay in k will have a reduced expected volunteering probability, and the volunteering probability of the player who is relocated is reduced from p/m_h to p/(m_k + 1). This shows that incumbent members of a team benefit from additional members. The new members disperse the probability of contributing. The expected cost is placed on a larger number of shoulders. For an individual player who is moved from team h to team k, the expected contribution burden is smaller if team k plus its new member is larger than the team which this player left.

Comparing partitions of different size

Sections 2.2 to 2.3 studied the volunteering game for given partitions of a set of players into teams. When comparing partitions in which the partition with fewer teams has at least two teams, the following applies:

Proposition 3

Consider two partitions: $P_{K}$ with K teams and $P_{K - 1}$ with K − 1 ≥ 2 teams. (i) The sum of the expected payoffs of all m + 1 players is higher in the partition $P_{K - 1}$ than in the $P_{K}$ . (ii) Let j_k be a member of a team of size m_k in the partition $P_{K}$ and a member of a team of size m_k
′ ≥ m_k in the partition $P_{K - 1}$ . Then the expected payoff of j_k is higher in the partition $P_{K - 1}$ than in the partition $P_{K}$ .

Proof

Consider the first property (i). There are exactly K players in $P_{K}$ that contribute with a positive probability — exactly one from each team. Also, there are exactly K − 1 players in $P_{K - 1}$ that contribute with a positive probability — again, exactly one from each team. According to Proposition 2, these players all have an expected equilibrium payoff of 1 − c. In $P_{K}$ there are (m + 1) − K players that never contribute. Each of them has the expected equilibrium payoff equal to 1 − c^K/(K−1). Similarly, $P_{K - 1}$ has (m + 1) − (K − 1) players that never contribute, and each of them has an expected equilibrium payoff equal to 1 − c⁽^{K−1)/(K−2)} > 1 − c^K/(K−1). Never contributing players are more numerous in $P_{K - 1}$ than in $P_{K}$ and each of them has a higher expected payoff than strict non-contributors in $P_{K}$ , whereas players who contribute with a positive probability in the equilibrium all have a lower expected payoff of 1 − c and are more numerous in $P_{K}$ than in $P_{K - 1}$ . The assertion (i) now follows from summing these effects. In algebraic terms, the sum of the expected payoffs in $P_{K - 1}$ exceeds that in $P_{K}$ by

(m + 1 - (K - 1)) ((1 - c^{\frac{K - 1}{K - 2}}) - (1 - c^{\frac{K}{K - 1}})) + ((1 - c) - (1 - c^{\frac{K}{K - 1}})) .

(11)

Consider next (ii). A blindly drawn player who is a member of a team of size m_k in the partition $P_{K}$ has an expected payoff of (1/m_k)(1 − c) + ((m_k − 1)/m_k)(1 − c^K/(K−1)). A representative player as a member of a team of size m_k_′ in the partition $P_{K - 1}$ has an expected payoff of (1/m_k′)(1 − c) + ((m_k_′ − 1)/m_k′)(1 − c^{(K−1)/(K−2)}). If m_k_′ > m_k, then such a player from team k′ has a larger probability of being a strict non-contributor than such a player from team k. As all players that become players who end up in the role as player who contributes with positive probability have a lower payoff equal to (1 − c) < (1 − c^K/(K−1)) < (1 − c^{(K−1)/(K−2)}), this confirms (ii).

As a corollary of Proposition 3, we can compare a partition that emerges from another partition if two teams inside that partition unite/merge. It follows from Propositions 2 and 3 that all players are better off: players who are inside the teams that are united and players in bystanding teams.

Proposition 3 also has implications for stability, that is, for the question of whether team members like to be in a team or would prefer to leave their given team. After leaving the old team and constituting a solo player ”team”, this solo-player and all players would prefer this player to return to the old team. Hence, leaving the team would be disadvantageous for this player, and also for all other players.

The results suggest that players like to be in large teams and they also prefer a smaller number of teams. If the partition of the set of players is made by a designer who maximizes the sum of players’ payoffs, in the absence of other constraints and costs the solution would be one single coordinated team. It would lead to a first-best outcome for which the discrete public good would be provided with certainty and in which precisely one player volunteers and bears the cost of volunteering.

Accounting for the cost of coordination

The size of teams might hit limits based on technological, geographical or other factors. And even if larger teams are feasible, coordination in larger teams might be increasingly costly. For instance, successful coordination within the team might require bilateral communication channels between all its members. The number of such bilateral communication channels for a team of size m_k is (1/2)(m_k²−m_k). This number is overproportionally increasing in m_k. If each communication channel has a fixed cost, the strong increase in the number of such channels might be one illustrative reason why coordination might have costs that are positive, increasing and convex in the team’s size.

Suppose there is a decision-maker outside the set M of potential contributors who can choose a partition into teams and wants to maximize aggregate welfare, defined as the sum of the expected payoffs of the players in M minus the costs of providing the teams with internal coordination infrastructures. When searching for the optimal partition, such a decision-maker faces a trade-off between higher coordination benefits due to a smaller number of larger teams and the increasing costs of providing the means for coordination inside teams. To study this optimization problem more formally, let the total coordination costs for a team with m_k members have the following properties:

(A1): C(1) = 0;

(A2): C(m_k) is increasing and strictly convex in m_k;

(A3): C(m + 1) > mc.

Assumption (A1) sets the floor of coordination costs for small teams: A single-player team has zero costs of coordination. (A2) states that coordination costs increase more than linearly in the size of the team — C(m_k) is an increasing and convex function. Assumption (A3) is a sufficient condition to make sure that the coordination cost of a team that consists of all potential volunteers exceeds the benefits from coordinated public good provision compared to the Diekmann’s (1985) benchmark. It is more plausible that this assumption is fulfilled if m is large. We can ask: what are the properties of a partition that maximizes welfare?

Proposition 4

Under (A1), (A2) and (A3), any optimal partition of players into a given number K of teams has the property that all teams are almost the same size: Their sizes differ by at most one player.

Proof

For C(1) = 0 and C(m + 1) > mc, a team of size m + 1 has an overall sum of players’ payoffs of (m + 1) − c − C(m + 1) < (m + 1) (1 − c), where (m + 1) (1 − c) is the sum of payoffs for K = m + 1. The payoffs in the partition with solo players has a higher payoff than the grand team, which shows that K = m + 1 is suboptimal. Consider a given partition with 1 < K < m + 1 and team sizes m₁ ≥ … ≥ m_K. Under Proposition 2, the number of teams determines the total provision probability as well as the total expected volunteering costs. Accounting for b = 1, the expected public-good benefits net of volunteering costs are

\begin{matrix} B (K) & = & (m + 1) P - K p c \\ = & (m + 1) (1 - c^{\frac{K}{K - 1}}) - K (1 - \sqrt[K - 1]{c}) c . \end{matrix}

These expected benefits B(K) are independent of how the players are assigned to the K different teams. This reduces the optimization problem for a given K to the choice of team sizes. Let there be two teams k and k′ which differ in size by more than one player, that is, m_k ≥ m_k_′ + 2. It can be shown that this assignment is not optimal. Compare the coordination costs with those in a partition that is identical to the one considered in all but the two teams k and k′, but relocates one member of team k from this larger team to the smaller team k′. This reassignment keeps B(K) constant. Furthermore, as C(m_k) is increasing and strictly convex in m_k, it follows that C(m_k) + C(m_k_′) > (C(m_k − 1) + C(m_k_′ + 1)) if m_k ≥ m_k_′ + 2.

Proposition 4 shows that increasing and strictly convex coordination costs make equal team sizes optimal. Given $(1 + m) \in N$ and $K \in N$ , the term (m + 1)/K will often not be a natural number. In these cases it cannot be avoided that there are teams of different sizes. A direct implication of Proposition 4 is that, for an optimal choice of the partition, the number of players in any two teams cannot differ by more than one. Due to indivisibilities, the proof could not rely on calculus and first-order conditions. But the insight on the optimality of team sizes relative to each other can be used to narrow down the search to identify the optimal number and structure of teams. Consider (m + 1)/K and define ν((m + 1)/K) as the largest natural number smaller than (m + 1)/K. This means that possible partitions into K teams that obey the optimality property derived in Proposition 4 consist of K − ((m + 1) − Kν((m + 1)/K) teams of size ν((m + 1)/K) and (m + 1) − Kν((m + 1)/K) teams of size ν((m + 1)/K) + 1. Each of these partitions has the same expected total sum of payoffs and this payoff sum can be calculated using the results in Proposition 2, and so for all K = 2, 3, …, (m + 1). For each K this determines exactly one sum of players’ payoffs net of coordination costs and the optimal K is the one with the largest welfare among these.

Conclusion

As pointed out in a large literature and illustrated by the examples described in the Introduction, a task that is a public good to many players might be provided by single volunteers if the volunteer’s cost is smaller than the volunteer’s own benefit from the public good. However, such situations often suffer from coordination failure, leading to outcomes in which the public good is not provided at all, and situations arise of the overprovision in terms of duplication of volunteers’ efforts. Coordination could solve this problem, but full coordination or coordination among large subgroups might be too difficult a task. For instance, if coordination requires communication among all members of a group, then the required number of communication links between its members and the aggregate cost implied has quadratic growth in the number of team members.

Smaller subsets of players inside the set of all potential volunteers might be able to coordinate among themselves. But if there is no coordination between such teams, would coordination inside such teams improve the welfare outcome? How does the answer depend on the number of teams? And would players like to be members of such teams, or would they prefer to stand alone? Would standing alone give them better free-rider opportunities? As is shown in the analysis carried out in the present study, if the number of teams is smaller, then the problem of miscoordination is smaller, the probability of overall provision is higher and the problem of effort duplication is lower. Also, in expectation, being a member in one of the teams with several players is better than being a stand-alone player. Members of a team would not benefit from departing from the team. And a split of a given team into smaller teams harms all potential volunteers. They are interested in a situation with few large teams, and aim to be in a large team themselves.

The analysis is silent about what the specific governance is at work that might determine the number and size of teams or the allocation of single players between the teams. If the team structure is a matter of choice, then it must take into account that a coordination device might have a cost, and that a device that works in larger teams might be increasingly more costly in the size of the team. Partitions into a smaller number of larger teams generates higher welfare in terms of the public good outcome. However, the higher cost of coordination must be accounted for by someone. If these costs or difficulties increase more than proportionate in team size, this suggests that an intermediate number of teams of roughly equal sizes maximize overall welfare.

The formal model necessarily abstracts from many relevant aspects that arise in the different analyses of the volunteer’s dilemma. The set of possible volunteers might be heterogeneous with respect to their costs of volunteering or their benefits from the public good. Various types of uncertainty might be relevant. The timing of actions might follow a sequential order, and issues of observability and incomplete information might become relevant in a dynamic setting. Thinking about applications in biology, it could also be interesting to explore the optimality of team size and structure for equilibrium concepts that focus on evolutionary stability. The analysis in the paper concentrates on a specific question and reveals a surprising property: one might have assumed that coordination within teams would increase the likelihood of a contribution coming from a large team, or that individual players would benefit from better free-rider opportunities. Instead, given the nature of coordination inside teams, the probability that there is at least one team with a volunteer depends only on the number of teams, but not on their size or on the heterogeneity between different teams.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Kai A. Konrad

Notes

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Multiple teams and the volunteer’s dilemma

Abstract

Keywords

Introduction

The game-theory model

The formal framework

Equilibrium

Welfare

Comparing partitions of different size

Accounting for the cost of coordination

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Notes

References