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Maximin behavior and efficient allocation
211
Economics Letters 6 (1980) 21 I-215 North-Holland Publishing Company
MAXIMIN
BEHAVIOR
AND EFFICIENT
ALLOCATION
Vincent
P. CRAWFORD
Uniwrsity
of Culiforniu, Sun Diego, LA Jolla, CA 92093, USA
Received
3 February
*
1981
A mechanism that can implement any Pareto-efficient allocation in a two-person Edgeworth box economy when agents choose among uncertain alternatives by the maximin criterion is described. This, coupled with the association of maximin and its generalizations with complete ignorance, suggests that deliberately withholding information from agents about each other’s preferences might allow the design of mechanisms with improved performance.
1. Introduction An intriguing possibility in the design of resource-allocation mechanisms is that deliberately withholding information from agents might, because of its strategic effects, actually allow an improvement in performance. This Letter reports a preliminary investigation of that possibility. It describes a simple mechanism that always generates Pareto-efficient allocations in two-person pure-exchange economies when agents respond to uncertainty about each other’s preferences with maximin behavior. The mechanism is also unbiased, in Hurwicz’s sense of being capable of realizing, after suitable adjustment, any Pareto-efficient allocation. Further, it can be made to satisfy an appealing ordinal equity condition when agents are perceived to have equal rights to the bundle of goods to be allocated. The maximin criterion can be viewed as a special case of the Hurwicz a-criterion, axiomatized by Arrow and Hurwicz (1972). These criteria are
* I am grateful to D.V.T. Bear for a helpful discussion, and to Clifford B. Donn (1977) whose conjecture about a different mechanism led me to formulate the mechanism discussed here. Financial support from the National Science Foundation under Grant SES 79-05550 is also gratefully acknowledged.
01651765/81/0000-0000/$02.50
0 North-Holland
212
K P. Cruwford / Muximin
behavior md efficient ullocutmn
deemed most appropriate for choices in uncertain situations characterized by complete ignorance (which is formalized in Arrow and Hurwicz’s axiom system). They share with dominant-strategy Nash equilibrium the distinct advantage, for implementation purposes, that an agent’s optimal action is independent of other agents’ actions. Thus, if agents could be induced by the withholding of information to adopt maximin or the a-criterion, it might be possible to avoid some common criticisms of implementation achieved under other solution concepts without running afoul of the Gibbard- Satterthwaite impossibility result. Related work, which considers the problem for public goods under the assumption (stronger than the assumptions maintained here in the private-goods case) that preferences are quasilinear, is reported in Thomson (1979)
2. The mechanism The mechanism will be described informally, in the two-good case, for simplicity. Formalization and extension to the general case are not difficult. Let U be the set of all continuous and strictly increasing (but not necessarily quasiconcave or differentiable) two-good utility functions. Each agent has preferences representable by some utility function in b, and knows only that the other agent has some such preferences. Agents respond to this uncertainty by maximizing their minimum possible utilities. One agent is, perhaps randomly, designated Proposer (P), and the other is designated Chooser (C). In the mechanism, P proposes a number of alternative allocations and C chooses among them. The restrictions on P’s proposals must be chosen to ensure both Pareto-efficiency and unbiasedness. A limiting, continuous version of the mechanism, which achieves these goals literally, is described first. In this version, P must report a surface that satisfies all of the restrictions satisfied by the indifference surfaces of utility functions in U; in the two-good case, it must therefore be a continuous and strictly downward-sloping curve. The surface is further required to divide the volume of the Edgeworth box in the proportions d and l-d, with the ‘d ’ part lying ‘below’ the surface, on the side of P’s origin. This requirement is easily shown to be independent of the choice of units of measurement for goods.
V.P. Crawford / Maximin behavror and efficrent allocation
213
For any value of d, P’s maximin-optimal policy is to report his true indifference surface that just satisfies the constraint on volume proportions. Given my monotonicity assumption, this indifference surface is clearly unique, and allowing lower volume proportions would not alter P’s optimal policy. The optimality of this policy follows simply because if some parts of the reported surface were at higher utility levels than others, they could be ‘lowered’, and the extra volume thereby made available used to place a higher utility floor under the surface. The continuity requirement rules out obvious pathological exceptions to this. Since C faces no uncertainty, his maximin-optimal choice from the allocations on P’s proposed surface is utility-maximizing, and therefore guarantees Pareto-efficiency, as desired. Finally, it is plain that varying the parameter d between zero and unity allows any Pareto-efficient allocation to be achieved. Of course, the planner who controls d must generally have more information to implement a more specific welfare criterion; the situation encountered here is no different from that encountered with the competitive mechanism. None of these conclusions would be altered if the set U were restricted to contain only quasiconcave utility functions. A discrete form of the mechanism (suggested in this particular form by D.V.T. Bear) yields approximately Pareto-efficient allocations with much simpler strategy spaces. In the Bear mechanism, P makes a certain fixed number of proposals, equally spaced and appropriately intersecting the boundary of the Edgeworth box, and the integrals that define the volume proportions that appear in the continuous version are replaced with discrete approximations constructed from the proposals. The requirements on spacing and boundary intersection are necessary to ensure that P does not respond to the incentives to concentrate his proposals, which would destroy the mechanism’s Pareto-efficiency. This form of the mechanism should make it possible to obtain allocations that are nearly Pareto-efficient (ignoring the costs associated with complexity of the mechanism) with relatively few proposals.
3. Removing asymmetry While there is no substitute for the good judgment of the planner in situations where agents are in ethically different positions (perhaps because they have made different contributions to the bundle or have different initial wealths), something more can be said when they are
214
V.P. Crawford
/ Muxrmin
behavior
and efficient
allocution
perceived to have equal rights to the bundle to be allocated. In this case, the mechanism’s asymmetric treatment of agents might reasonably be viewed as a drawback. Thus, it is natural to try to choose a value of d that makes both agents content with their roles, perhaps in the sense (ordinal and independent of interpersonal comparisons) that neither would prefer the other’s. This approach proved successful in removing the asymmetry across roles in the ‘equal-division divide-and-choose’ procedure discussed in Crawford (1980); see Crawford (1979) for the details. It turns out that a case can be built on these grounds for setting d= l/2. At first, this may seem a bit surprising, since there is no guarantee in general that any value of d will make both agents content with their roles; and if there were such a value, it seems most likely that there would be an infinity of them. Further, experimenting with diagrams makes it clear that the allocation an agent actually gets as Chooser when d = l/2 always yields him utility at least as great as, and generally greater than, his maximin-optimal policy as Proposer with d = l/2. This is an immediate consequence of the ‘unused’ volume between agents’ indifference surfaces. In spite of this, d = l/2 is the unique value that makes both agents indifferent ex ante between the two roles. To see this, note that when P is Proposer for a given value of d, the worst outcome for C is when P proposes one of C’s indifference curves. (This cannot happen in general, and the argument is no longer valid, if U is restricted to contain only quasiconcave utility functions.) Because the proportion of volume ‘below’ C’s indifference surface is l-d, his security level as Chooser when the Proposer is allowed volume proportion d is exactly equal to his security level as Proposer when he is allowed volume proportion l-d. These security levels are equal if and only if d = l/2, and the same conclusion holds for P by symmetry. There is, nevertheless, an important grain of truth in the argument at the end of the second-to-last paragraph. If agents adopt a lexicographic maximin rule rather than simple maximin, both will prefer the Chooser’s role when d = l/2: both roles have the same security level, but the Chooser’s is almost sure to yield higher utility. But both agents will also prefer the Proposer’s role as soon as d rises above l/2, no matter how little. So in a sense, complete removal of asymmetry is not really possible in this case.
V. P. Crawford / Maximn
behavior and efficient allocutum
215
4. Conclusion While the generalization of the above results to the m-private-goods case is immediate, it is not apparent how to generalize to other environments, or to replace maximin with the less conservative a-criterion. (Thomson has obtained some partly positive results for the a-criterion for public goods under the quasilinearity assumption.) Thus, while the positive results obtained here suggest that something might be learned from further investigations in this direction, it seems likely that the mechanisms that emerge, if any, will look different from the one described here.
References Arrow, Kenneth J. and Leonid Hurwicz, 1972, An optimality criterion for decision-making under ignorance, in: CF. Carter and J.L. Ford, eds., Uncertainty and expectations in economics: Essays in honor of G.L.S. Shackle (Kelley, Clifton, NJ) I- 11. Crawford, Vincent P., 1979, A procedure for generating Pareto-efficient egalitarianequivalent allocations, Econometrica 47, 49-60. Crawford, Vincent P., 1980, A self-administered solution of the bargaining problem, Review of Economic Studies 47, 385-392. Donn, Clifford B., 1977, Games final-offer arbitrators might play, Industrial Relations 16. 306-314. Thomson, William, 1979, Maximin strategies and elicitation of preferences, in: J.-J. Laffont. ed., Aggregation and revelation of preferences (North-Holland, Amsterdam) 245-268.
Economics Letters 6 (1980) 21 I-215 North-Holland Publishing Company
MAXIMIN
BEHAVIOR
AND EFFICIENT
ALLOCATION
Vincent
P. CRAWFORD
Uniwrsity
of Culiforniu, Sun Diego, LA Jolla, CA 92093, USA
Received
3 February
*
1981
A mechanism that can implement any Pareto-efficient allocation in a two-person Edgeworth box economy when agents choose among uncertain alternatives by the maximin criterion is described. This, coupled with the association of maximin and its generalizations with complete ignorance, suggests that deliberately withholding information from agents about each other’s preferences might allow the design of mechanisms with improved performance.
1. Introduction An intriguing possibility in the design of resource-allocation mechanisms is that deliberately withholding information from agents might, because of its strategic effects, actually allow an improvement in performance. This Letter reports a preliminary investigation of that possibility. It describes a simple mechanism that always generates Pareto-efficient allocations in two-person pure-exchange economies when agents respond to uncertainty about each other’s preferences with maximin behavior. The mechanism is also unbiased, in Hurwicz’s sense of being capable of realizing, after suitable adjustment, any Pareto-efficient allocation. Further, it can be made to satisfy an appealing ordinal equity condition when agents are perceived to have equal rights to the bundle of goods to be allocated. The maximin criterion can be viewed as a special case of the Hurwicz a-criterion, axiomatized by Arrow and Hurwicz (1972). These criteria are
* I am grateful to D.V.T. Bear for a helpful discussion, and to Clifford B. Donn (1977) whose conjecture about a different mechanism led me to formulate the mechanism discussed here. Financial support from the National Science Foundation under Grant SES 79-05550 is also gratefully acknowledged.
01651765/81/0000-0000/$02.50
0 North-Holland
212
K P. Cruwford / Muximin
behavior md efficient ullocutmn
deemed most appropriate for choices in uncertain situations characterized by complete ignorance (which is formalized in Arrow and Hurwicz’s axiom system). They share with dominant-strategy Nash equilibrium the distinct advantage, for implementation purposes, that an agent’s optimal action is independent of other agents’ actions. Thus, if agents could be induced by the withholding of information to adopt maximin or the a-criterion, it might be possible to avoid some common criticisms of implementation achieved under other solution concepts without running afoul of the Gibbard- Satterthwaite impossibility result. Related work, which considers the problem for public goods under the assumption (stronger than the assumptions maintained here in the private-goods case) that preferences are quasilinear, is reported in Thomson (1979)
2. The mechanism The mechanism will be described informally, in the two-good case, for simplicity. Formalization and extension to the general case are not difficult. Let U be the set of all continuous and strictly increasing (but not necessarily quasiconcave or differentiable) two-good utility functions. Each agent has preferences representable by some utility function in b, and knows only that the other agent has some such preferences. Agents respond to this uncertainty by maximizing their minimum possible utilities. One agent is, perhaps randomly, designated Proposer (P), and the other is designated Chooser (C). In the mechanism, P proposes a number of alternative allocations and C chooses among them. The restrictions on P’s proposals must be chosen to ensure both Pareto-efficiency and unbiasedness. A limiting, continuous version of the mechanism, which achieves these goals literally, is described first. In this version, P must report a surface that satisfies all of the restrictions satisfied by the indifference surfaces of utility functions in U; in the two-good case, it must therefore be a continuous and strictly downward-sloping curve. The surface is further required to divide the volume of the Edgeworth box in the proportions d and l-d, with the ‘d ’ part lying ‘below’ the surface, on the side of P’s origin. This requirement is easily shown to be independent of the choice of units of measurement for goods.
V.P. Crawford / Maximin behavror and efficrent allocation
213
For any value of d, P’s maximin-optimal policy is to report his true indifference surface that just satisfies the constraint on volume proportions. Given my monotonicity assumption, this indifference surface is clearly unique, and allowing lower volume proportions would not alter P’s optimal policy. The optimality of this policy follows simply because if some parts of the reported surface were at higher utility levels than others, they could be ‘lowered’, and the extra volume thereby made available used to place a higher utility floor under the surface. The continuity requirement rules out obvious pathological exceptions to this. Since C faces no uncertainty, his maximin-optimal choice from the allocations on P’s proposed surface is utility-maximizing, and therefore guarantees Pareto-efficiency, as desired. Finally, it is plain that varying the parameter d between zero and unity allows any Pareto-efficient allocation to be achieved. Of course, the planner who controls d must generally have more information to implement a more specific welfare criterion; the situation encountered here is no different from that encountered with the competitive mechanism. None of these conclusions would be altered if the set U were restricted to contain only quasiconcave utility functions. A discrete form of the mechanism (suggested in this particular form by D.V.T. Bear) yields approximately Pareto-efficient allocations with much simpler strategy spaces. In the Bear mechanism, P makes a certain fixed number of proposals, equally spaced and appropriately intersecting the boundary of the Edgeworth box, and the integrals that define the volume proportions that appear in the continuous version are replaced with discrete approximations constructed from the proposals. The requirements on spacing and boundary intersection are necessary to ensure that P does not respond to the incentives to concentrate his proposals, which would destroy the mechanism’s Pareto-efficiency. This form of the mechanism should make it possible to obtain allocations that are nearly Pareto-efficient (ignoring the costs associated with complexity of the mechanism) with relatively few proposals.
3. Removing asymmetry While there is no substitute for the good judgment of the planner in situations where agents are in ethically different positions (perhaps because they have made different contributions to the bundle or have different initial wealths), something more can be said when they are
214
V.P. Crawford
/ Muxrmin
behavior
and efficient
allocution
perceived to have equal rights to the bundle to be allocated. In this case, the mechanism’s asymmetric treatment of agents might reasonably be viewed as a drawback. Thus, it is natural to try to choose a value of d that makes both agents content with their roles, perhaps in the sense (ordinal and independent of interpersonal comparisons) that neither would prefer the other’s. This approach proved successful in removing the asymmetry across roles in the ‘equal-division divide-and-choose’ procedure discussed in Crawford (1980); see Crawford (1979) for the details. It turns out that a case can be built on these grounds for setting d= l/2. At first, this may seem a bit surprising, since there is no guarantee in general that any value of d will make both agents content with their roles; and if there were such a value, it seems most likely that there would be an infinity of them. Further, experimenting with diagrams makes it clear that the allocation an agent actually gets as Chooser when d = l/2 always yields him utility at least as great as, and generally greater than, his maximin-optimal policy as Proposer with d = l/2. This is an immediate consequence of the ‘unused’ volume between agents’ indifference surfaces. In spite of this, d = l/2 is the unique value that makes both agents indifferent ex ante between the two roles. To see this, note that when P is Proposer for a given value of d, the worst outcome for C is when P proposes one of C’s indifference curves. (This cannot happen in general, and the argument is no longer valid, if U is restricted to contain only quasiconcave utility functions.) Because the proportion of volume ‘below’ C’s indifference surface is l-d, his security level as Chooser when the Proposer is allowed volume proportion d is exactly equal to his security level as Proposer when he is allowed volume proportion l-d. These security levels are equal if and only if d = l/2, and the same conclusion holds for P by symmetry. There is, nevertheless, an important grain of truth in the argument at the end of the second-to-last paragraph. If agents adopt a lexicographic maximin rule rather than simple maximin, both will prefer the Chooser’s role when d = l/2: both roles have the same security level, but the Chooser’s is almost sure to yield higher utility. But both agents will also prefer the Proposer’s role as soon as d rises above l/2, no matter how little. So in a sense, complete removal of asymmetry is not really possible in this case.
V. P. Crawford / Maximn
behavior and efficient allocutum
215
4. Conclusion While the generalization of the above results to the m-private-goods case is immediate, it is not apparent how to generalize to other environments, or to replace maximin with the less conservative a-criterion. (Thomson has obtained some partly positive results for the a-criterion for public goods under the quasilinearity assumption.) Thus, while the positive results obtained here suggest that something might be learned from further investigations in this direction, it seems likely that the mechanisms that emerge, if any, will look different from the one described here.
References Arrow, Kenneth J. and Leonid Hurwicz, 1972, An optimality criterion for decision-making under ignorance, in: CF. Carter and J.L. Ford, eds., Uncertainty and expectations in economics: Essays in honor of G.L.S. Shackle (Kelley, Clifton, NJ) I- 11. Crawford, Vincent P., 1979, A procedure for generating Pareto-efficient egalitarianequivalent allocations, Econometrica 47, 49-60. Crawford, Vincent P., 1980, A self-administered solution of the bargaining problem, Review of Economic Studies 47, 385-392. Donn, Clifford B., 1977, Games final-offer arbitrators might play, Industrial Relations 16. 306-314. Thomson, William, 1979, Maximin strategies and elicitation of preferences, in: J.-J. Laffont. ed., Aggregation and revelation of preferences (North-Holland, Amsterdam) 245-268.