Mechanism design: how to implement social goals

Transcription

1 Mechanism Design Mechanism design: how to implement social goals From article by Eric S. Maskin Theory of mechanism design can be thought of as engineering side of economic theory Most theoretical work focuses on existing economic institutions and tries to explain or forecast outcomes of these institutions. Mechanism design theory reverses the direction of inquiry: identify desired outcomes (social goals) and ask whether institutions could be designed to attain the goal Outcomes, goals and mechanisms Outcome: depends on context For public good: quantities provided (intercity highways, environmental protection) For electorate: choice of candidate Auctioneer: allocation of assets across buyers and the payments made Home buyer: specification of characteristics and builder remuneration Desirability or optimality: depends on context Public goods: net social surplus maximization Electorate: candidate wins head- to- head opponents Mechanism: is an institution, procedure, or game for determining outcomes Public goods: method of determining levels of goods Auction: auction format Political elections: electoral procedure House building: design of contract Subject is interesting when the government (or auctioneer) will typically not have the information for optimal allocation, etc. Mechanism designers must thus generate the information needed as outcomes are executed As individuals do have the information, and have their own objectives, the mechanism must be incentive compatible 1

2 Three basic questions of MD 1. When is it possible to design incentive- compatible mechanisms for attaining social goals? 2. What form might these mechanism take when they exist? 3. When is finding such mechanisms ruled out theoretically? An example Two consumers of energy, Alice and Bob An energy authority is charged with choosing the type of energy to be used by Alice and Bob Options from which single selection must be made: gas, oil, nuclear power, and coal Suppose two possible states of the world State 1: consumers place relatively little weight on future State 2: attach great importance to future (rates of discount low) An example Preferences for Alice and Bob Alice cares primarily about convenience when it comes to energy (lower ranking energy source becomes either messier or more cumbersome) In state 2 her rankings change as she anticipates technical advances will make gas, coal and especially nuclear to become easier to use Bob is interested particularly in safety. So in state 1 where present has greatest weight he favors nuclear over oil, oil over coal, coal over gas In state 2, nuclear wastes problem looms large but improvement on oil and gas safety expected 2

3 Social choice rule Energy authority is interested in selecting an energy source that both consumers are reasonably happy with, e.g., first or second choice, so that in state 1 oil is the optimal choice while in state 2 gas is the optimal choice Social choice rule prescribes oil in state 1 and gas in state 2 (Table 2) Authority does not know state Could ask each consumer to announce the state and choose oil if both said state 1 and gas if both said state 2 and flip a coin if a mixed response results But notice that Alice has an incentive to say state 2 regardless of the actual state or what Bob says (prefers oil in both states). She increases the likelihood of her preferred outcome whatever Bob says. Similarly Bob will say state 1. Taken together the outcome is a randomization between oil and gas, i.e., only a 50% chance of an optimal outcome Let us suppose that the authority has the consumers participate in the mechanism of Table 3. Alice chooses Top or Bottom as her strategy; simultaneously, Bob chooses Left or Right as his strategy. The outcome of those choices is given by the matrix: In state 1 the Nash equilibrium is also optimal Observe that in state 1 Left is a dominant strategy for Bob Given Bob s dominant strategy (Left), Alice is better off choosing Top so (Top, Left) is a unique Nash eq 3

4 In state 2 the Nash equilibrium is also optimal Observe that in state 2 Bottom is a dominant strategy for Alice Given Alice s dominant strategy, Bob is better off choosing Right so (Bottom, Right) is a unique Nash eq Table 3 Mechanism properties Achieves the optimal outcome even though i. The mechanism designer does not even know the actual state, and ii. Alice and Bob are interested only in their own preferences, not those of the authority Because Table 3 Nash equilibrium mechanism outcomes coincide with optimal outcomes in each state, we say that the mechanism implementsthe authority s social choice rule in Nash equilibrium Brief history of mechanism design The intellectual history of mechanism design theory goes back at least to nineteenth- century utopian socialists such as Robert Owen and Charles Fourier. Repulsed by what they viewed as the evils of the burgeoning capitalist system, these thinkers argued that socialism offered a more humane alternative and sometimes became involved in setting up experimental communities such as New Harmony, Indiana. Brief history of mechanism design More direct influence comes from the Planning Controversy 1930s (Oskar Lange and Abba Lerner) Lange and Lerner believed that done right planning could replicate the performance of free markets (and even correct serious market failures like those on display with the Great Depression) Friedrich von Hayek and Ludwig von Mises staunchly denied the possibility of that a planned systems could even approach free market success 4

5 Brief history of mechanism design Leo Hurwicz found the controversy frustrating: arguments incomplete in part because of lack of technical apparatus (game theory and mathematical programming) Hurwicz and others produced broad consensus that the market is indeed the best mechanism if 1. No market concentration of market power 2. No significant externalities When these conditions don t hold mechanisms can generally improve the market Two branches: 1. special structured settings (how to design auctions, allocate public goods, structure contracts) 2. General and abstract analysis Implementation of social choice rules Three central questions rephrased: 1. Under what conditions can a social choice rule be implemented? 2. What form does an implementation mechanism take? 3. Which social choice rules cannot be implemented? Monotonicity Monotonicity is a key property to implementability of in Nash equilibrium Suppose that outcome a is optimal in state θ according to social choice rule f, that is, f θ = a. Then if a doesn t fall in anyone s ranking relative to any other alternative in going from state θ to state θ &, monotonicity requires that a also be optimal in state θ &, that is θ & : f θ = a. However, if a doesfall relative to some outcome b in someone s ranking, monotonicity imposes no restriction. Monotonicity satisfied in energy example Oil is optimal in state 1. In going from state 1 to 2 oil falls in Alice s ranking, relative to both coal and nuclear power. The fact that gas not oil is optimal in state 2 means there is no violation of monotonicity. Similarly for Bob and gas Therefore, the scr satisfies monotonicity and thus possibility of implementing it. 5

6 A violation of monotonicity: consider a different preference pattern Observe that although oil is optimal in state 1, it is not optimal in state 2, despite the fact that it falls in neither Alice s nor Bob s rankings between states 1 and 2 Hence we can conclude that there is no mechanism that implements the scr of Table 4. Theorem 1 (Maskin 1977) Theorem 1: If a social choice rule is implementable, then it must be monotonic Suppose to the contrary that the scr in Table 4 were an implementing mechanism Then the mechanism would necessarily contain a pair of strategies (s,, s. ) that result in outcome oil and constitute a Nash equilibrium in state 1. Maksin claims that (s,, s. ) must also constitute a Nash equilibrium in state 2. To see this.. Contradiction Note first that Bob has no incentive to deviate unilaterally from s. in state 2, since (i) he has no such incentive in state 1 (by definition of Nash eq) and (ii) his preference ranking is the same in both states. Contradiction Furthermore, Alice has no incentive to deviate from s, in state 2. To see this, observe that if, contrary to the claim, Alice gained from deviating unilaterally from s, in state 2, she must thereby be inducing the outcome gas (because this is the only outcome she prefers to oil in state 2). But Alice also prefers gas to oil in state 1, and so would benefit from the same deviation in that state, contradicting the assumption that (s,,s. ) constitutes a Nash equilibrium in state 1. 6

7 Theorem 2 Hence, (s,,s. ) is indeed a a Nash equilibrium in state 2. But the outcome it generates oil is not optimal in that state, establishing that the scr is not implementable after all. No veto power: if all individuals, except possibly one, agree that a particular outcome is best (all put a on top of their ranking) the remaining individual cannot veto it. Theorem 2 (Maskin 1977): Suppose that there are at least three individuals. If the social choice rule satisfies monotonicity and no veto power, then it is implementable. 7