Anchoring on Utopia: a generalization of the Kalai–Smorodinsky solution

Abstract

Many bargaining solutions anchor on disagreement, allocating gains with respect to the worst-case scenario. We propose here a solution anchoring on utopia (the ideal, maximal aspirations for all agents), but yielding feasible allocations for any number of agents. The negotiated aspirations solution proposes the best allocation in the direction of utopia starting at an endogenous reference point which depends on both the utopia point and bargaining power. The Kalai–Smorodinsky solution becomes a particular case if (and only if) the reference point lies on the line from utopia to disagreement. We provide a characterization for the two-agent case relying only on standard axioms or natural restrictions thereof: strong Pareto optimality, scale invariance, restricted monotonicity, and restricted concavity. A characterization for the general (n-agent) case is obtained by relaxing Pareto optimality and adding the (standard) axiom of restricted contraction independence, plus the minimal condition that utopia should be selected if available.

Appendix: Properties of the sets \(V_t\)

For each $t\in [1,n]$ let $V_t$ be given by (1). The following lemma summarizes the geometric properties of these sets as required in the proofs in the main text.

Lemma 1

For every $t\in [1,n]$,

• (a) $m(V_t)=(1,\ldots ,1)$;

• (b) $SPO(V_t)=\left\{ x\in V_t\;\left| \; \; \sum _{i=1}^n x_i =t\right. \right\}$;

• (c) $\mathrm{WPO}(V_t)=SPO(V_t) \cup \left\{ x \in V_t\;\left| \; \; x_j=1 \text { for some } j\in \{1,\ldots ,n\}\right. \right\}$; and

• (d) $V_t = \left( \frac{n-t}{n-1}\right) V_1 + \left( 1-\frac{n-t}{n-1}\right) V_n$.

Proof

(a), (b), and (c) are straightforward. To see (d), we proceed by double inclusion. Let $x\in V_1$, $y\in V_n$, and $z=\left( \frac{n-t}{n-1}\right) x + \left( 1-\frac{n-t}{n-1}\right) y$. Then $z\in [0,1]^n$ and $\sum _{i=1}^n z_i= \left( \frac{n-t}{n-1}\right) \sum _{i=1}^n x_i + \left( 1-\frac{n-t}{n-1}\right) \sum _{i=1}^n y_i \le \left( \frac{n-t}{n-1}\right) (1) + \left( 1-\frac{n-t}{n-1}\right) (n) = t$. Hence, $y\in V_t$.

Let $z\in V_t$. Note that $y=(1,\ldots ,1)\in V_n$ and define x by $x_i=\frac{(n-1)z_i-t+1}{n-t}$. Then, $\sum _{i=1}^n x_i= \frac{1}{n-t}\left( (n-1) \sum _{i=1}^n z_i + n(1-t)\right) \le \frac{1}{n-t}\left( (n-1)t + n(1-t)\right) =1$ and $x\in V_1$. Note that $\left( \frac{n-t}{n-1}\right) x_i + \left( 1-\frac{n-t}{n-1}\right) y_i = \frac{n-t}{n-1} \left( \frac{(n-1)z_i-t+1}{n-t}\right) + \left( 1-\frac{n-t}{n-1}\right) = \frac{1}{n-1} ( (n-1)z_i ) = z_i$. Hence, $z\in \left( \frac{n-t}{n-1}\right) V_1 + \left( 1-\frac{n-t}{n-1}\right) V_n$. $\square$

The next result identifies two useful properties of bargaining solutions satisfying restricted concavity, when applied to the sets $V_t$.

Lemma 2

Let f be a bargaining solution for n-agent problems satisfying weak Pareto optimality, restricted concavity, and utopia fulfillment. Then, for every $t\in [1,n]$,

• (a) $f(V_t)= \left( \frac{n-t}{n-1}\right) f(V_1) + \left( 1-\frac{n-t}{n-1}\right) f(V_n)$, and

• (b) $f(V_t)\in SPO(V_t)$.

Proof

(a) By Lemma 1(a), $m(V_1)=m(V_n)$. Let $z=\left( \frac{n-t}{n-1}\right) f(V_1) + \left( 1-\frac{n-t}{n-1}\right) f(V_n)$. By weak Pareto optimality, $f(V_1)\in \mathrm{WPO}(V_1)$ and by Lemma 1(b,c), $SPO(V1)=\mathrm{WPO}(V1)$. Hence, by Lemma 1(b), $\sum _{i=1}^n f_i(V_1)=1$. By utopia fulfillment, $f(V_n)=(1,\ldots ,1)$. We obtain that $\sum _{i=1}^n z_i =t$. By restricted concavity, $f(V_t)\ge z$. Since $\sum _{i=1}^n f_i(V_t)\le t$, it follows that $f(V_t)=z$.

(b) Follows from part (a) and Lemma 1(b). $\square$