The maximin equilibrium and the PBE under ambiguity

Abstract

This note refers to the recent work on ambiguous implementation by de Castro–Liu–Yannelis (Econ Theory 63:233–261, 2017). The authors discuss, under condition of ambiguity, the implementation as maximin equilibria of maximin individually rational and ex ante maximin efficient allocations. An explicit example is used to support their analysis. We analyse further the example used by de Castro–Liu–Yannelis (2017). We show that in the formulated game tree the proposed allocation is implementable through a backward induction argument. Also it is shown that a perfect Bayesian equilibrium (PBE) exists, leading to different allocations. Comparisons are drawn between the maximin and the PBE implementations. We consider also briefly the meaning of the incentive compatibility (IC) of proposed allocations.

Appendix

In order to check the individual rationality of the proposed allocation, we consider Agent 1. We want to solve the following:

Problem 1

$$\begin{aligned} \text {Minimize }p_1\sqrt{5}+p_2\sqrt{4.8}\quad \text {subject to }p_1+p_2=2/3. \end{aligned}$$

It is straightforward that we obtain the corner solution $p_1=0$ and $p_2=2/3$. This justifies the formula used by de Castro et al.:

$$\begin{aligned} {\frac{2}{3}}\text {min}\{\sqrt{5}, \sqrt{5}\}+{\frac{1}{3}}\sqrt{1}=1.824<{\frac{2}{3}}\text {min}\{\sqrt{5}, \sqrt{4.8}\} +{\frac{1}{3}}\sqrt{1.2}=1.826. \end{aligned}$$

Similarly we obtain the identical relation for Agent 2. Hence the proposed feasible allocation is individually rational and Definition 2 is satisfied. Each agent loses 0.2 and gains 0.2 units of different goods but from distinct starting quantities and the agent becomes strictly better off.

Next we want to show that the proposed allocation above is ex ante maximin efficient. First, we want to give a justification of the choice of the proposed allocation. One can trace the steps through a social welfare function. Alternatively we can start the analysis from Problem 3.

However it is of some interest to see that the proposed allocation emerges also as the one that maximizes a social welfare function, W, in which the two agents have equal weights. We are looking for the solution of:

Problem 2

$$\begin{aligned} \begin{array}{ll} \text {Maximize }&{}W=2{\sqrt{x_1(a)}}+2{\sqrt{x_1(b)}}+{\sqrt{x_1(c)}}+2{\sqrt{x_2(a)}}+{\sqrt{x_2(b)}}+2{\sqrt{x_2(c)}}\\ \text {Subject to }&{}x_1(a)+x_2(a)=10,\quad x_1(b)+x_2(b)=6,\quad x_1(c)+x_2(c)=6, \end{array} \end{aligned}$$

where the coefficients 2 and 1 are suggested by ${\frac{2}{3}}$ and ${\frac{1}{3}}$ of the de Castro et al. formula above.

The problem is separable into the sub-problems per individual constraint, as there is no transferability of endowments between periods. We have the unique solutions $x_1(a)=x_2(a)=5$, $x_1(b)=4.8,x_2(b)=1.2$ and $x_1(c)=1.2,x_2(c)=4.8$. There does not exist another feasible allocation which both agents prefer and at least one of them prefers strictly. It follows that we also have the unique solution of:

Problem 3

$$\begin{aligned} \begin{array}{ll} \text {Maximize }&{}W''=2{\sqrt{x_1(b)}}+{\sqrt{x_1(c)}}+{\sqrt{x_2(b)}}+2{\sqrt{x_2(c)}}\\ \text {Subject to }&{} x_1(b)+x_2(b)=6,\quad x_1(c)+x_2(c)=6. \end{array} \end{aligned}$$

The solution is $X_1=(x_1(b), x_1(c))=(4.8, 1.2)$ and $X_2=(x_2(b), x_2(c))=(1.2, 4.8)$; and $u_1=1.826$ and $u_1=1.826$.

But of course for Definition 3 we have to go to the maximin criterion of Definition 1.

We must take into account that we have priors implying, as shown in Problem 1, multiplication of a minimum by 2/3. We know that for Problem $W''$ we have obtained unique solutions $X_1$ and $X_2$. Suppose we attach to these the quantities $y_1=y_2=5$. Then we get the solution of Problem 2 which is also what was called above the proposed allocation.

We shall show that this resulting allocation is ex ante maximin efficient. We argue as follows. Suppose one increases $y_1$ and reduces $y_2$, only. The utility of Agent 1 cannot increase because min$(y_1,4.8)$ stays at 4.8 and the utility of Agent 2 can only be reduced if for example min$(y_2,4.8)=y_2$ which will now itself carry the coefficient 2/3.

Analogously, looking at it from the point of view of Agent 2, if we only decrease $y_1$ and increase $y_2$, we do not obtain a superior allocation according to the maximin criterion.

Next, suppose that we only change the vectors $X_1$ and $X_2$. The uniqueness of the solution of Problem 3 implies that the utility value of at least one of the $X_i$s has been reduced. Without loss of generality, let this be that of $X_1$.

In all circumstances, there will be, according to the maximin criterion, a reduction in the utility for Agent 1. Therefore changes in $X_1$ and $X_2$ alone cannot lead to a strictly preferred allocation.

Suppose now we consider a combination of all changes at the same time. Suppose, without loss of generality, that the utility value of $X_1$ has been reduced. No matter what the change in $y_{1}$ is, the utility of Agent 1 according to the maximin criterion will be reduced. Hence the proposed allocation is ex ante maximin efficient.

Now we turn our attention to the question of whether the proposed allocation is the unique one which has this property. Consider the allocation

$$\begin{aligned} x= \left( \begin{array}{c@{\quad }c@{\quad }c} x_1{(a)} &{} x_1{(b)}&{} x_1{(c)} \\ x_2{(a)} &{} x_2{(b)}&{} x_2{(c)} \end{array}\right) = \left( \begin{array}{c@{\quad }c@{\quad }c} 5-\epsilon &{} 4.8 &{} 1.2 \\ 5+\epsilon &{} 1.2 &{}4.8 \end{array}\right) , \end{aligned}$$

where $\epsilon$ is very small. Notice that $\epsilon$ and $-\epsilon$ are chosen to preserve feasibility because we do not assume free disposal. Also $\epsilon$ is chosen small enough so that we still have min$(5-\epsilon , 4.8)=4.8$, i.e. $-0.2<\epsilon <0.2.$

The new allocation is individually rational and this is easy to see. Analogously to the previous proof above we obtain, for Agent 1 and Agent 2, respectively,

$$\begin{aligned} {\frac{2}{3}}\text {min}\{\sqrt{5}, \sqrt{5}\}+{\frac{1}{3}}\sqrt{1}< & {} {\frac{2}{3}}\text {min}\{\sqrt{5-\epsilon }, \sqrt{4.8}\} +{\frac{1}{3}}\sqrt{1.2},\quad {\text {and}}\\ {\frac{2}{3}}\text {min}\{\sqrt{5}, \sqrt{5}\}+{\frac{1}{3}}\sqrt{1}< & {} {\frac{2}{3}}\text {min}\{\sqrt{5+\epsilon }, \sqrt{4.8}\} +{\frac{1}{3}}\sqrt{1.2}. \end{aligned}$$

Next, we have to show that, in spite of the new $(y'_1, y'_2)=(x_1(a),x_2(a))$, any such allocation is ex ante maximin efficient. We repeat the previous argument.

Suppose now considering simultaneous changes of the allocation vectors, that $X_1$ has been reduced. No matter what the change in $y'_{1}$ is, the utility of Agent 1 according to the maximin criterion will be reduced. Hence the proposed allocation is ex ante maximin efficient.