Incentives of agency theory – Borda, Bowen, Vickrey: Incentives in Voting

Since the beginning of the theory on voting, the issue of strategic voting was noticed. Borda (1781) recognized it when he proposed his famous Borda rule:

My scheme is only intended for honest men.

We have to wait for Bowen (1943) to see a first attempt at addressing the issue of strategic voting. For allocating public goods, Bowen (as we mentioned in section 1.3) was searching in voting for an alternative to the missing expression of preferences that exists in markets for private goods. He realized the difficulty of strategic voting:

At first thought it might be supposed that this information could be obtained from his vote……………….. But the individual could not vote intelligently, unless he knew in advance the cost to him of various amounts of the social good, and in any case the results of voting would be unreliable if the individual suspected that his expression of preference would influ- ence the amount of cost to be assessed against him.

—Bowen (1943, p. 45)

Bowen assumed that the distribution of the cost of the public good was exoge- nously fixed (e.g., equal sharing of cost) and considered successive votes on incre- ments of the public good. He observed that at each step it is in the interest of each voter to vote yes or no according to his true preferences. Such a procedure leads to the optimal level of public good if agents are myopic and consider only their incentives at each step.11 Black (1948), years after Borda, Condorcet, Laplace, and Dogson, reconsidered the theory of voting and exhibited a wide class of cases (single-peaked preferences) for which majority voting leads to the transitivity of social choice, a solution to the 1785 Condorcet paradox. Black eliminated, by assumption, strategic issues:

When a member values the motions before a committee in a definite order, it is reasonable to assume that, when these motions are put against each other, he votes in accordance with his valuation.

—Black (1948, p. 134), cited in Arrow and Scitovsky (1969)

When Arrow (1951) founded the formal theory of social choice by proving that there is no “reasonable” voting method yielding a nondictatorial social tran- sitive ranking of social alternatives when no restriction is placed on individual preferences, he also abstracted from the gaming issues and noticed that

[t]he point here, broadly speaking, is that, once a machinery for making social choices from individual tastes is established, individuals will find it profitable, from a rational point of view, to misrepresent their tastes by their actions or, more usually, because some other individual will be made so much better off by the first individual’s misrepresentation that he could compensate the first individual in such a way that both are better off than if everyone really acted in direct accordance with his tastes.”12

—Arrow (1951, p. 7)

In a paper that provides a very lucid exposition of Arrow’s impossibility the- orem, Vickrey (1960) raised the question of strategic misrepresentation of pref- erences in a social welfare function that associates a social ranking to individual preferences:

There is another objection to such welfare functions, however, which is that they are vulnerable to strategy. By this is meant that individuals may be able to gain by reporting a preference differing from that which they actually hold.

—Vickrey (1960, p. 517)


Such a strategy could, of course, lead to a counterstrategy, and the process of arriving at a social decision could readily turn into a “game” in the technical sense.

—Vickrey (1960, p. 518)

Dummett and Farquharson (1961) would indeed pursue the analysis of such voting games in terms of noncooperative Nash equilibria. Vickrey (1960) further explained that the social welfare functions that satisfy the assumptions of Arrow’s theorem, in particular the independence assumption, are immune to strategy. Then comes his conjecture, acknowledged by Gibbard (1973):

It can be plausibly conjectured that the converse is also true, that is, that if a function is to be immune to strategy and to be defined over a com- prehensive range of admissible rankings, it must satisfy the independence criterion, though it is not quite so easy to provide a formal proof of this.

—Vickrey (1960, p. 588)

Therefore, Vickrey is led, through Arrow’s theorem, to an impossibility result, namely the nonexistence of any method of aggregating individual preferences or of any voting mechanism that is nonmanipulable. The route toward the impossibility of nonmanipulable and nondictatorial mechanisms via Arrow’s theorem was sug- gested. A complete proof, the greatest achievement of social choice theory since Arrow’s theorem, came thirteen years later in Gibbard (1973).13 The importance of Gibbard’s theorem for incentive theory lies in showing that with no prior knowl- edge of preferences, nondictatorial collective decision methods cannot be found where truthful behavior is a dominant strategy. The positive results of incentive methods in practice will have to be looked for in restrictions on preferences, as in the principal-agent theory, or in the relaxation of the required strength of incentives by giving up dominant strategy implementation.

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.

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