We will study this issue in the context of social choice functions (rather than social welfare functions as we have considered so far). However, the challenge is that we don't know the values, but rather each player knows their value, and we want to make sure that our mechanism decides the allocation - the social choice - in a way that cannot be strategically manipulated.
Clarke Pivot Rule
We need to show that for playeriwith valuation vi, the utility when we declare vi is not less than the utility when we declare vi. As mentioned, the Clarke pivot rule does not fit many situations where valuations are negative; i.e. when alternatives have costs for the players. With the Clarke pivot rule, the players actually always pay money to the mechanism, while the natural interpretation in the case of costs would be the opposite.
The spirit of the Clarke pivot rule in such cases can be captured by a modified rule that chooses to maximize social welfare.
Examples .1 Auction of a Single Item
Maximizing social welfare means that the items are allocated to the highest bidders, and in the VCG mechanism with the pivot rule, each of them must pay the highest offered price. The project has a publicly known costC and is valued by every citizen (a privately known) valuevi. Usually we think thatvi ≥0, but the case of allowing vi <0, i.e. citizens being harmed by the project, is also included.).
This is technically not a sub-case of our definition of social welfare maximization, since our definition assumed no cost or value to the planner, but it becomes so by adding an additional actor "government" whose valuation space is a single-male valuation, giving the cost C of carrying out the project and 0 otherwise.). Suppose we want to obtain a communication path between two specific nodes, t ∈V; i.e. the set of alternatives is the set of all possible-t paths in G and player e has the value 0 if the chosen path does not contain and the value -ce if the chosen path does. This fits the spirit of the rotation rule, since the "without" mechanism simply cannot use containing paths.
Implementation in Dominant Strategies
Games with Strict Incomplete Information
The entire behavior of the player in such a setting is captured by a function that determines which actionx to take for each possible type - this is called a strategy. Thus, the notion of ex-post Nash requires that si(ti) is the best response to si(t−i) for each possible value of t−i, i.e. without knowing anything about t−i, but only knowing other players' strategy-i forms as functions. The notion of a dominant strategy requires that si(ti) is the best answer to any possible x−i, i.e. without knowing anything about t−i or abouts−i.
Both of these definitions seem too good to be true: how likely is it that a player has a single action that is a best response to allx−ior even to alls−i(t−i). However, in the context of Mechanism Design - where we can design the game - we can sometimes check that they do exist. While the idea of dominant strategy equilibrium at first glance seems much stronger than ex-post Nash, this is only due to actions that are never used.
Mechanisms
In normal cases, you wouldn't expect games with strictly incomplete information to have any of these equilibria. Formally, we can specify for each player a set of possible actionsXi, an outcome functiona:X1× · · · ×Xn→A that chooses an alternative in A for each action profile, and payment functions p:X1× · · · × each player specifies for each action profile. Now the players are placed in a game with strictly incomplete information and we can expect them to reach an equilibrium point (if one exists).
A stronger requirement would be that all equilibria have this property, or even stronger, that only one unique equilibrium point exists.
The Revelation Principle
Thus, in particular, this is true for allx−i =s−i(t−i) and any xi =si(ti), which gives the definition of incentive compatibility of the mechanism (f, p1, . . . , pn). Furthermore, the payoffs of the players in the incentive-compatible mechanism are the same as the payoffs obtained in the equilibrium of the original mechanism. In particular, general mechanisms can be adaptive (multi-round), which significantly reduces the communication (or computational) burden on the players or the auctioneer compared to an inflexible direct mechanism.
Characterizations of Incentive Compatible Mechanisms In Section 9.3 we saw how to implement the most natural social choice function: maxi-
- Direct Characterization
- Weak Monotonicity
- Weighted VCG
- Single-Parameter Domains
- Uniqueness of Prices
- Randomized Mechanisms
An example is the case of combinatorial auctions studied in Chapter 11. 9.5 Characterizations of incentive-compatible mechanisms In section 9.3 we saw how to implement the most natural social choice function: maxi-. that approaches the maximization of social welfare, but is different from it. In the remainder of the section, we will provide characterizations of when such mechanisms are incentive compatible. A mechanism is incentive compatible if and only if it satisfies the following conditions for everyiand everyv−i: i) The payment pi does not depend on vi, but only on the chosen alternative f(vi, v−i). ii) The mechanism optimizes for each player.
It turns out that when the domain of preferences is unrestricted, the only mechanisms compatible with incentives are simple variations on the VCG mechanism. It is not difficult to see that any incentive-compatible mechanism can easily be converted into a normalized mechanism. Thus, it is sufficient to characterize normalized mechanisms. The only mechanisms that are compatible with incentives and that maximize social welfare are those involving VCG payments. ii).
In the bilateral trade problem (section 9.3.5.3), the only incentive mechanism that maximizes social welfare and makes no payments in the event of no trade is the one shown there that subsidizes trade. A normalized randomized mechanism in a single parameter domain is expectedly compatible with incentives if and only if we have that for everyi and every fixed v−i. i) the functionwi(vi, v−i) is monotonically non-decreasing inviand and (ii) pi(vi, v−i)=vi·w(vi, v−i)−vi.
Bayesian–Nash Implementation
Bayesian–Nash Equilibrium
We must explicitly emphasize that the randomization in a randomized mechanism is completely controlled by the mechanism designer and has nothing to do with any distributional assumptions about player valuations, as will be discussed in the next section. The mechanism implements the social choice function f :T1× · · · ×Tn→ A in the Bayesian sense, if for some Bayesian-Nash equilibrium s1,. As in the case of implementations of the dominant strategy, Bayesian implementations can be transformed into ones that are true in the Bayesian sense.
Proposition 9.44 (Revelation Principle) If there exists an arbitrary mechanism that implements f in the Bayesian sense, then there exists a truthful mechanism that implements f in the Bayesian sense. Moreover, the expected payments of the players in the truthful mechanism are identical to those obtained in equilibrium, in the original mechanism. The proof is similar to the proof of the same principle in the dominant-strategy setting given in Proposition 9.25.
First Price Auction
In general, however, the Bayesian performance depends on the Di distributions, and there are many cases where a Bayesian-Nash equilibrium exists even though no dominant strategy exists. Let us now see how this situation fits into the Bayesian–Nash setting described above: The type space TAlice Alice and TBobof Boba are nonnegative real numbers, with tAlice denoted by a and tBob by b. Even for this very simple first-price auction, the answer is not clear for the general DAlice and DBob distributions.
This means that the first-price auction also maximizes social welfare, as does the second-price auction. Alice's utility is 0 if she loses and a−x if she wins and paysx, so her expected utility from bidx is given by uAlice=P r[Alice wins with bidx]·(a−x), where the probability over the prior distribution to b. Therefore, to optimize the value of x, we need to find the maximum of the function 2x(a−x) in the range 0≤x≤1/2.
Revenue Equivalence
We will derive a formula for the expected payment for this type that depends only on the expected payment for type ti0 and on the social choice function f. Note that under these notations the expected utility of playeri with typetiλ declaring tiλ is given by the expression vλ·wλ−pλ. Similarly, a player with type tiλ+ prefers to report the truth rather than tiλ, so we have vλ+·wλ−pλ≤vλ+·wλ+−pλ+.
But if we are willing to change the social choice function, then we can certainly increase the income. Set a reserve price of 1/2, then sell to the highest bidder for a price that is the maximum of the low bid and the reserve price, 1/2. Then a quick calculation will reveal that the expected revenue in this auction is 5/12, which is more than the 1/3 achieved in the regular second price or first price auctions.
Further Models
- Risk Aversion
- Interdependent Values
- Complete Information Models
- Hidden Actions
Roughly speaking, in such models each player receives a signal that is positively correlated (in a strong technical sense called affiliation) not only with his own value, but also with the values of other players. In such settings, increasing English auctions are "better" (generate more revenue) than the non-adaptive second-price auction (which is equivalent to an English auction in private value models): while As bidding progresses, each bidder receives information from the other bidder that increases his estimate of his value. A prototypical example is that of King Solomon: two women, each claiming the baby as her own.
There are several implementation concepts in such a setting, and in general, designing a mechanism in this setting is much easier. All mechanism design theory tries to overcome the problem that players have private information unknown to the mechanism designer. A different stumbling block appears in many settings: players can perform hidden actions that are not visible to the "mechanism". This complementary problem to the private information problem has been extensively studied in economics and has recently begun to be addressed in computer science settings.
Notes
The positive results in Mechanism Design in the quasi-linear setting originate from the seminal work of Vickrey (1961), who particularly studied single-item auctions and multi-unit auctions with downward valuations. The general framework of mechanism design and its basic concepts have developed in microeconomic theory mostly in the 1970s, and mostly in the general Bayesian approach, which we will only come to in section 9.6. The uniqueness of pricing is the analogue of the revenue equivalence theorem in a Bayesian setting, which is due to Myerson (1981); Green and Laffont (1977) showed that in the dominant strategy of welfare maximization of social choice functions.
The consequence of the impossibility of balanced bilateral trade is highlighted in Myerson and Satterthwaite (1983) in the Bayesian setting. As previously mentioned, most of the development in Mechanism Design took place in this setting. The revenue equivalence theorem, the form of expected payment in single-parameter domains, as well as an analysis of revenue-maximizing auctions, come from Myerson (1981).
