Subgame Perfect Coalition Formation

Abstract

We analyze a dynamic game where players can each make offers to other players to form coalitions. We show that these games have a unique subgame perfect equilibrium outcome that is individually rational and, when players can make enough proposals, Pareto optimal. We also provide sufficient conditions for equilibrium to implement core coalition structures.

A: Additional Proofs

Details of Example 2

Suppose that all of the above coalitions are feasible, and that the order of proposers is $O=(1,2,3,4,5,6)$. For example, if 1’s offer is rejected, 2 makes an offer. If that gets rejected, then 3 makes an offer, and so on. We now derive the subgame perfect Nash equilibrium outcome of this game (which turns out to be unique, as we show later).

• 1 Consider any subgame in which player 6 makes an offer. Clearly, every offer will be accepted, since rejection implies that the player who rejects an offer becomes a singleton (and each player in our example prefers to be on a coalition with anyone to being by themselves).

• 2 Consider a subgame in which players 1–4 have all been rejected, and it is player 5’s turn to make an offer. If any offer by 5 is rejected at this point, the outcome will be $\{\{1\},\{2\},\{3\},\{4\},\{5,6\}\}$, since 5 is 6’s most preferred coalitionmate, and by the preceding logic. Consequently, any offer by 5 to coalitions with players 1–4 will be accepted. Since 5 most prefers 2, who is still available, this is the offer 5 will make, and it will be accepted. Moreover, since 1 is the most preferred remaining player by 6, the outcome in this subgame is $\{\{1,6\},\{2,5\},\{3\},\{4\}\}$. SPNE outcome in this subgame: $\{\{1,6\},\{2,5\},\{3\},\{4\}\}$.

• 3 Consider a subgame in which players 1–3 have all been rejected, and it’s player 4’s turn to make an offer. If player 4 makes an offer to $\{3,4\}$, the coalition $\{3,4\}$ will form if 3 accepts or coalitions $\{3\},\{4\}$ will form if it rejects (from subgame (2) above). Since 4 prefers 3 to any others, he can do no better than making an offer to $\{3,4\}$, with the outcome being $\{\{1,6\},\{2,5\},\{3,4\}\}$. It is thus an equilibrium of this subgame for 4 to offer to $\{3,4\}$, and for 3 to accept. SPNE outcome in this subgame: $\{\{1,6\},\{2,5\},\{3,4\}\}$.

• 4 Consider a subgame in which players 1 and 2 have been rejected, and now it’s player 3’s turn. If player 3 makes an offer to $\{2,3\}$, 2 prefers to reject, because 2 prefers to be with 5 (the outcome of subgame (3)) than with 3. If player 3 makes an offer to $\{3,4\}$, this offer is accepted, and the outcome is again $\{\{1,6\}, \{2,5\},\{3,4\}\}$. Making any other offer cannot improve 3’s utility. SPNE outcome in this subgame: $\{\{1,6\}, \{2,5\},\{3,4\}\}$.

• 5 Consider a subgame in which player 1 was rejected, and player 2 now makes an offer. If 2 makes an offer to $\{1,2\}$, 1 will accept, because if 1 rejects, they end up paired with 6 (subgame (4)), and 1 prefers being with 2. Since 1 is the most preferred pick by 2, 2 would strictly prefer making this offer to any other. Thus, coalition $\{1,2\}$ will form. Once this happens, $\{3,4\}$ will coalition up since they are then conditional soulmates, which implies that $\{5,6\}$ will coalition up as well. SPNE outcome in this subgame: $\{\{1,2\},\{3,4\},\{5,6\}\}$.

• 6 Now, consider player 1’s options. If 1 makes an offer to $\{1,3\}$ or $\{1,4\}$, it will be rejected, because both 3 and 4 prefer to be with each other than to be with 1 (and they end up together if they reject 1). If 1 makes an offer to $\{1,5\}$, 5 will accept, since 5 prefers to be with 1 than to be with 6 (which is the outcome if 5 rejects 1’s offer). Consequently, 1 will make an offer to $\{1,5\}$ in equilibrium, and 5 will accept, forming the coalition $\{1,5\}$. Now, by the time 2 gets to move, 1 and 5 are off the market. Suppose that 2 and 3 then make offers which are rejected. If 4 then makes an offer to $\{3,4\}$, 3 will accept, because otherwise both will end up by themselves (since 6 will make an offer to $\{2,6\}$). Since 3 accepts, the coalitions $\{3,4\}$ and $\{2,6\}$ form in this subgame, with the resulting SPNE outcome in this subgame being $\{\{1,5\},\{3,4\},\{2,6\}\}$. Backing up, suppose it’s 3’s turn to make an offer. If 3 offers to $\{2,3\}$, 2 will accept, because otherwise 2 ends up with 6. Since 2 is 3’s most preferred player, the coalition $\{2,3\}$ will then form. Consequently, the SPNE of the subgame in which 2 is rejected after 1 and 5 coalition up is $\{\{1,5\},\{2,3\},\{4,6\}\}$. Finally, suppose that 2 makes an offer to $\{2,4\}$, its most preferred remaining coalitionmate. 4 will then accept, since rejecting the offer will cause 4 to be coalitioned up with 6, who is less preferred than 2. Consequently, the coalitions $\{2,4\}$ and $\{3,6\}$ will form. This means that the following outcome is a SPNE outcome of the full game: $\{\{1,5\},\{2,4\},\{3,6 \}\}$.

Proof of Lemma 1

We prove this by induction, after noting that $A_{R}$ is unique by Remark 1.

Base Case: Suppose that the coalition T has been proposed. Consider an arbitrary sequential order of accept/reject decisions for players in T. Suppose that i is last in that order and all players before i have accepted. Then, i will clearly accept since for any $\pi _{R}\in \Pi _{R}$, by assumption either $T\succ _{i}\pi _{R,i}$ or, if $T=\pi _{R,i}$ and, from lexicographic time preferences this holds even if, in a further subgame, another proposer proposes T and it is accepted.

Inductive Step: Consider a player i such that none of the players $k<i$ in the accept/reject order have rejected. Our inductive hypothesis is that if i accepts, then in every SPNE of the residual T-subgame all players $k^{\prime }>i$ (which follow i in the order) accept. It is immediate that i’s unique optimal strategy is then to accept, since for any $\pi _{R}\in \Pi _{R}$ either $T\succ _{i}\pi _{R,i}$, or $T=\pi _{R,i}$ , and acceptance is preferred by lexicographic time preferences. The final step is to observe that when i is the first player in the order, none of the players before i have rejected, because no one precedes i. $\square$

Proof of Theorem 1

We prove this by showing the result for a subgame with only one remaining proposer and then appealing to backward induction.

Base Case: Consider an arbitrary subgame with only one player, i, who can still make a proposal and the set of feasible coalitions for i,  denoted by $\mathcal {T}_{i}$ (none of the others matter). We show that in this subgame in every SPNE all proposals are accepted and result in a unique outcome. First, define $\mathcal {T}_{i}^{IR}=\{T\in \mathcal {T}_{i}|T\succ _{j}\{j\}\forall \ j\in T\}\cup \{i\}$, that is, a subset of feasible coalitions in which every coalition is preferred by all its members over being by themselves unioned with $\{i\}$. Clearly, every coalition offer $T\in \mathcal {T}_{i}^{IR}$ other than $\{i\}$ will be accepted. Let $T_{i}^{*}$ be i’s most preferred coalition in $\mathcal {T}_{i}^{IR}$. If $T_{i}^{*}=\{i\}$, by lexicographic preferences i strictly prefers to propose to and to accept coalition$\ \{i\}$. Otherwise, $T_{i}^{*}\succ _{i}\{i\}$. Because all $j\in T_{i}^{*}$ accept and form a coalition, coalitions which have been formed thus far are fixed, and any remaining players become singletons, the subgame has a unique SPNE outcome.

Now consider the player who is the next to last proposer. Standard backward induction for extensive games with perfect information can now be applied and the above result holds for the “rolled back” game. This can be continued until the first player in the ordering O is to make an offer, which proves the result.¹⁵$\square$