## Abstract

Elements of manipulation are strategic/sophisticated voting, and agenda setting. Strategic voting results into the Gibbard-Satterthwaite theorem (Gibbard) which is about a specific kind of decision mechanism for social choice functions–a function that specifies for each combination for possible preference orderings. The theorem states that “for any non- dictatorial and non-imposed social choice function, there is vulnerability to manipulation.” This means that expressing one’s true preferences is not a dominant strategy always; rather, there are preference profiles where at least one of the individuals has an incentive not to express his/her ‘true’ preferences. Also, it can be stated that the only social choice function that is free from manipulation and not vulnerable to manipulation in any way, is dictatorship. This is what I have set to elaborate in this annotated bibliography.

Alan Gibbard, ‘Manipulation of Voting Schemes: A General Result’ Econometrica 41 (1973): 587-601

The author is a University Professor of Philosophy at the University of Michigan. He specializes in ethical theory and his research centers on the claims that the concept of meaning is a normative concept. In regard to this topic, Allan Gibbard stipulates that “all voting schemes are manipulable” This same opinion is proved in Dummet & Farquharson’s Stability in Voting (1961).

Definition of Ordering

An ordering of Z is two-place relation P such that for all

Also,

Definition of Voting Scheme

Voting scheme is defined as a function which assigns a member of set of alternatives (Z) to each possible preference n-tuple (P1, P2,…., Pn) for a given n and set Z.

Where n represents the voters, Z is the set of alternatives, Pi the orderings of Z for each voter i

Manipulation of the voting scheme results when one misrepresents his/her preferences and secures an outcome preferred to the “honest outcome” (Gibbard 1973, p.587). Manipulation is only meaningful when the honest preferences are known. The main result is that any non-dictatorial voting scheme with at least 3 possible outcomes is subject to individual manipulation (Gibbard 1973, p. 587)

A game form

Any scheme that makes the outcome to depend on the individual action of some specified form or strategies is referred to as the game form (Gibbard 1973, p.587).

Let X be the set of possible outcomes, n the number of players, and Si for each player i, a set of strategies for i

A game form can then be described by the function

The function takes each possible strategy n-tuple {S1, …., Sn} with to an outcome

Voting scheme and Game form

Voting scheme is a special case of game form. Every non-chance procedure by which individual choices of contingency plans for action determine an outcome is characterized by a game form. A game form does not specify what an `honest’ strategy would be, so there is no such thing as manipulability.

Manipulability: This is a game form property and n functions which takes each possible preference ordering to a strategy

For every individual K and preference ordering P,

is the K strategy representing P.

Now,

Dominant Strategy

If whatever anyone else does, a goal is achieved at least as the alternative strategy, then; the strategy is regarded as dominant (Gibbard 1973, p.587).

Let the strategy n-tuple be

Also, let

if the Kth term is replaced by t

If for every n-tuples s,

then the strategy t is predominant for k.

If for k and P, the strategy P is dominant for k, then the game form is considered to be straightforward.

Player K is considered a dictator if for every outcome of a game g, then

whenever

If there is a dictator for g, then the game form g is dictatorial.

Every straightforward game form that has at least three possible outcomes is dictatorial.

Mark Satterthwaite, ‘Strategy Proofness and Arrow’s Conditions’ Journal of Economic Theory 10 (1975): 187-217.

This article is about the existence and correspondence theorems for voting procedures and social welfares functions. The author, Mark (Allen) Satterthwaite, is a PhD holder in Economics from University of Wisconsin, Madison. His areas of expertise includes competition in healthcare, healthcare management, strategy, voting systems, Strategy-Proofness and Arrow’s Conditions from his PhD thesis The Existence of a Strategy Proof Voting Procedure: A Topic in Social Choice Theory (1973).

Definition of Strategy-proofness

A voting procedure Vnm at ballot set B= (B1, …, Bn) can be manipulated by an individual

If and only if there is a ballot Bi such that

The voting procedure is manipulable at ballot set B= (B1, …, Bn) if there is an individual

that can manipulate Vnm at B.

Nicholas R. Miller, “Election Inversions by the U.S. Electoral College” For presentation at the 2011 Annual Meeting of the Public Choice Society San Antonio, Texas March 10-13, 2011

According to Nicholas, An election inversion occurs when the candidate (or party) that wins the most votes (highest number of votes) from the nationwide electorate fails to win the most electoral votes (or parliamentary seats) and therefore loses the election. Such situation is also called ‘reversal of winners,’ ‘referendum paradox,’ ‘election reversal,’ ‘compound majority paradox,’ or ‘wrong winner.’

Electoral College Election Inversions

Manifest EC Election Inversions

Election

EC Winner

EC Loser EC Loser’s 2-P PV%

2000

271 [Bush (R)]

267 [Gore (D)]

50.27%

1888

233 [Harrison (R)] 168 [Cleveland (D)]

50.41%

1876

185 [Hayes (R)]

184 [Tilden (D)]

51.53%

The Probability of Election Inversions: First Cut – Historical Estimate

Number of Inversions/Number of elections (since 1828)

3/46 = .06522

Clearly, an important determinant of the probability of an election inversion is the probability of a close division of the popular vote. Considering only elections in which the winner’s popular vote margin was no greater than about 3 percentage points, the frequency of inversions is considerably higher.

The PVEV Step Function

The Full PVEV for 1988 Appears To Go Through the Perfect-Tie Point

Historical Magnitude and Direction of Election Inversion Intervals

The 25% vs. 75% Rule

Suppose we have k uniform districts each with n voters (both odd numbers).

A candidate can win by carrying a bare majority of (k + 1) / 2 districts, each with a bare majority of (n + 1) / 2 votes. Thus, a candidate can win with as few as

[(k + 1) / 2] × [(n + 1) / 2] = (nk + n + k +1) / 4 efficiently distributed total votes.

With n = 3 and k = 3, the last expression is 4/9 = 44.4%, but as n and k become large, the last expression approaches a limit of nk / 4, i.e., 25% of the total popular vote.

## Works Cited

Alan Gibbard, ‘Manipulation of Voting Schemes: A General Result’ Econometrica 41 (1973): 587-601

Mark Satterthwaite, ‘Strategy Proofness and Arrow’s Conditions’ Journal of Economic Theory 10 (1975): 187-217.

Keith Dowding, “In Praise of Manipulation” Government Department, London School of Economics and Political Science, Houghton Street, London.

Nicholas R. Miller, “Election Inversions by the U.S. Electoral College” For presentation at the 2010 Annual Meeting of the Public Choice Society San Antonio, Texas March 10-13, 2010