(In)efficiency and equitability of equilibrium outcomes in a family of bargaining games

Abstract

We construct a parametric family of (modified) divide-the-dollar games: when there is excess demand, some portion of the dollar may disappear and the remaining portion is distributed in a bankruptcy problem. In two extremes, this game family captures the standard divide-the-dollar game of Nash (Econometrica 21:128–140, 1953) (when the whole dollar vanishes) and the game studied in Ashlagi et al. (Math Soc Sci 63:228–233, 2012) (when the whole dollar remains) as special cases. We first show that in all interior members of our game family, all Nash equilibria are inefficient under the proportional rule if there are ‘too many’ players in the game. Moreover, in any interior member of the game family, the inefficiency increases as the number of players increases, and the whole surplus vanishes as the number of players goes to infinity. On the other hand, we show that any bankruptcy rule that satisfies certain normatively appealing axioms induces a unique and efficient Nash equilibrium in which everyone demands and receives an equal share of the dollar. The constrained equal awards rule is one such rule.

Appendix

This Appendix shows that, in Corollary 4.4, none of our axioms is implied by the other two axioms.

$\diamond$ A rule that satisfies CDI and EDCFC, but not FCCSEM: Let ${\dot{R}}$ be a weighted gains rule for some vector of positive weights, $w = \left( w_1, \ldots , w_n\right)$ (see Moulin 2000). As such, for any bankruptcy problem (c, E), we have ${\dot{R}}_i(c,E) = \min \{\lambda w_i, c_i\}$ for every $i \in N$ where $\lambda$ is such that $\sum _{i \in N} {{\dot{R}}_i (c,E)} = E$.

The rule ${\dot{R}}$ satisfies CDI, because any agent $i \in N$ with ${\dot{R}}_i(c,E) < c_i$ collects an award of $\lambda w_i$. After a decrease in his claim to some $c'_i \ge {\dot{R}}_i(c,E)$, it can be seen that $\min \{\lambda w_i, c'_i\} = \lambda w_i$, so that his award does not change. The rule ${\dot{R}}$ also satisfies FCCSEM, because any increase in the endowment amount increases $\lambda$, so that all unsatisfied agents become better-off.

However, ${\dot{R}}$ does not satisfy EDCFC for the weights vector $w = (1,2,3,4)$. For that, consider an example with $N= \{1,2,3,4\}$, $c=(20, 40, 60, 80)$, and $E = 100$. Then, $\lambda$ is found to be 10, so that the awards vector is ${\dot{R}}(c,E) = (10,20,30,40)$. But now, although $c_1 = 20 < 25 = \frac{100}{4} = \frac{E}{n}$, we have ${\dot{R}}_1(c,E) = 10 < c_1$.

$\diamond$ A rule that satisfies CDI and EDCFC, but not FCCSEM: Let ${\bar{R}}$ be defined as follows. In any given bankruptcy problem (c, E), let $i^* \in N$ be the agent who has the maximum claim among all agents $i \in N$ with $c_i \le \frac{E}{n}$. The rule ${\bar{R}}$ awards $c_i$ to any agent $i \in N$ with $c_i \le c_{i^*}$ and awards $c_{i^*}$ to any agent j with $c_j > c_{i^*}$. If there is some excess surplus, then it is given to the remaining agents lexicographically following an order of their index numbers. That is, the remaining surplus is given to agent 1 until his award reaches his claim; if there is still some excess surplus, it is given to agent 2 until his award reaches his claim; and so on.

The rule ${\bar{R}}$ satisfies CDI, because any agent $i \in N$ with ${\bar{R}}_i(c,E) < c_i$ needs to have $c_i > c_{i^*}$, so that after he collects $c_{i^*}$, the rest of his award was given according to his index number. Given that he is unsatisfied, a decrease in his claim to some $c'_i \ge {\bar{R}}_i(c,E)$ cannot change his award. The rule ${\bar{R}}$ also satisfies EDCFC, by definition.

However, ${\bar{R}}$ does not satisfy FCCSEM. For that, consider an example with $N= \{1,2,3,4\}$, $c=(10, 60, 20, 90)$, and $E = 100$. Then, from the first part of the rule definition, we know that $i^* = 3$, so that agents collect (10, 20, 20, 20) at first. The excess surplus is 30. It cannot be given to agent 1 since he is already fully compensated. It is then given to agent 2, and since there is no surplus left, the process stops here. We find that ${\bar{R}}(c,E) = (10, 50, 20, 20)$. Now, if we increase the surplus to $E' = 105$, following similar steps, we can find ${\bar{R}}(c,E') = (10, 55, 20, 20)$. But then, although ${\bar{R}}_4(c,E) < c_4$, we have ${\bar{R}}_4(c,E') = {\bar{R}}_4(c,E)$.

$\diamond$ A rule that satisfies FCCSEM and EDCFC, but not CDI: Let ${\tilde{R}}$ be defined as follows. In any given bankruptcy problem (c, E), let $i^* \in N$ be the agent who has the maximum claim among all agents $i \in N$ with $c_i \le \frac{E}{n}$. The rule ${\tilde{R}}$ awards $c_i$ to any agent $i \in N$ with $c_i \le c_{i^*}$ and awards $c_{i^*}$ to any agent j with $c_j > c_{i^*}$. If there is some excess surplus, then it is given to the latter type of agents by using the proportional rule.

The rule ${\tilde{R}}$ satisfies FCCSEM, because any agent $i \in N$ with ${\tilde{R}}_i(c,E) < c_i$ needs to have $c_i > c_{i^*}$, so that after he collects $c_{i^*}$, the rest of his award was given using the proportional rule, and it is known that ${\mathcal {P}}$ satisfies strict endowment monotonicty (see Thomson 2019), which implies FCCSEM. The rule ${\tilde{R}}$ also satisfies EDCFC, by definition.

However, ${\tilde{R}}$ does not satisfy CDI. For that, consider an example with $N= \{1,2,3,4\}$, $c=(10, 60, 20, 90)$, and $E = 100$. Then, from the first part of the rule definition, we know that $i^* = 3$, so that agents collect (10, 20, 20, 20) at first. The excess surplus is 30, which is then divided between agents 2 and 4 using the proportional rule. Given their claims, they additionally get the 2/5th and 3/5th of that excess surplus, respectively. As such, we find that ${\tilde{R}}(c,E) = (10, 32, 20, 38)$. Now, if we decrease agent 4’s claim to $c'_4 = 60$, we can find ${\tilde{R}}(c',E) = (10, 35, 20, 35)$, so that the awards vector changes following an allowed decrease in claims.