Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result Allan Gibbard - Manipulation of voting The result for game forms schemes: a general result (1973) Proof of theorem Conclusions Discussion Charlotte Vlek Literature June 23, 2009
Table of contents Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background Background The main result The result for game forms The main result Proof of theorem Conclusions The result for game forms Discussion Literature Proof of theorem Conclusions Discussion
Allan Gibbard Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result ◮ Allan Gibbard (1942 - ) The result for game forms ◮ University Professor of Philosophy at University of Proof of theorem Michigan Conclusions “My field of specialization is ethical theory” Discussion Literature “My current research centers on claims that the concept of meaning is a normative concept” (www-personal.umich.edu/ ∼ gibbard/)
Situation in 1973 Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background Conjectured: all voting schemes are manipulable. The main result The result for game ◮ Dummet & Farquharson: Stability in voting (1961) forms “it seems unlikely that there is any voting Proof of theorem Conclusions procedure in which it can never be Discussion advantageous for any voter to vote Literature “strategically”, i.e., non-sincerely.” (D.& F. 1961, p.34 in: Gibbard 1973, p.588) ◮ They prove a similar result but only for “majority games”, not for all voting schemes
Situation at the time Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game forms ◮ Vickrey: Utility, strategy and social decision rules Proof of theorem (1960): Conclusions ◮ IIA & positive association imply non-manipulability Discussion ◮ conjectured: non-manipulability implies IIA & PA. Literature Gibbard confirms Vickrey
Definitions - ordering Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game An ordering of Z is two-place relation P such that for all forms Proof of theorem x , y , z ∈ Z : Conclusions ◮ ¬ ( xPy ∧ yPx ) (totality) Discussion (logically equivalent to yRx ∨ xRy ) Literature ◮ xPz → ( xPy ∨ yPz ) (transitivity) (logically equivalent to ( zRy ∧ yRx ) → zRx )
Definitions - voting scheme Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game forms ◮ n voters Proof of theorem ◮ Z set of alternatives Conclusions ◮ P i orderings of Z for each voter i Discussion Literature A voting scheme is a function that assigns a member of Z to each possible preference n-tuple ( P 1 , P 2 , ..., P n ) for a given number n and set Z .
Definitions - manipulation Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game One manipulates the voting scheme if forms Proof of theorem “by misrepresenting his preferences, he secures Conclusions an outcome he prefers to the “honest” Discussion outcome” (Gibbard 1973, p.587) Literature Note that manipulation only has a meaning if we know the “honest” preferences too.
The main result Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game forms The main result Proof of theorem Conclusions “Any non-dictatorial voting scheme with at Discussion least 3 possible outcomes is subject to Literature individual manipulation” (Gibbard 1973, p. 587)
Definitions - Game form Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek “A game form is any scheme which makes an Background outcome depend on individual actions of some The main result specified sort, which I shall call strategies” (Gibbard 1973, p.587) The result for game forms Proof of theorem Formally: Conclusions ◮ X a set of possible outcomes Discussion ◮ n number of players Literature ◮ S i for each player i , a set of strategies for i . A game form is a function g : S 1 × S 2 × ... × S n → X that takes each possible strategy n-tuple � s 1 , ..., s n � with s i ∈ S i ∀ i to an outcome x ∈ X .
Voting scheme vs. Game form Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game ◮ Every non-chance procedure by which individual forms choices of contingency plans for action determine an Proof of theorem outcome is characterized by a game form Conclusions Discussion ◮ Voting scheme is a special case of game form Literature ◮ A game form does not specify what an ‘honest’ strategy would be, so there is no such thing as manipulability
Voting scheme vs. Game form Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result ◮ Manipulability is a property of a game form plus n The result for game forms functions σ k ( k ≤ n ) that take each possible Proof of theorem preference ordering to a strategy s ∈ S k . For each Conclusions individual k and preference ordering P , σ k ( P ) is the Discussion strategy for k which honestly represents P . Literature ◮ Now we have v ( P 1 , ..., P n ) = g ( σ 1 ( P 1 ) , ..., σ n ( P n ))
Definitions - dominant strategy Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek “A strategy is dominant if whatever anyone else Background does, it achieves his goals at least as well as would The main result any alternative strategy” (Gibbard 1973, p.587) The result for game forms Formally: Proof of theorem Conclusions ◮ let s = � s 1 , ..., s n � be a strategy n -tuple Discussion ◮ let s k / t = � s 1 , ..., s k − 1 , t , s k +1 , ..., s n � (replace k th Literature strategy by t ) A strategy t is P -dominant for k if for every strategy n -tuple s , g ( s k / t ) Rg ( s ). A game form is straightforward if for every individual k and preference ordering P , there is a strategy P -dominant for k .
Definitions - dictatorship Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game forms ◮ A player k is a dictator for a game form g if for Proof of theorem every outcome x there is a strategy s ( x ) for k such Conclusions that g ( s ) = x whenever s k = s ( x ). Discussion Literature ◮ A game form g is dictatorial if there is a dictator for g .
The result for game forms Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game forms Proof of theorem The result for game forms : Conclusions Every straightforward game form with at least three Discussion possible outcomes is dictatorial. Literature
The result for game forms Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The main result The result for game forms The result for game forms : Proof of theorem Every straightforward game form with at least three Conclusions possible outcomes is dictatorial. Discussion Corollary : Literature Every voting scheme with at least three outcomes is either dictatorial or manipulable.
Proof of theorem Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek Background The result for game forms : The main result Every straightforward game form with at least three The result for game possible outcomes is dictatorial. forms Proof of theorem Proof: Conclusions ◮ Let g be a straightforward game form with at least 3 Discussion outcomes Literature ◮ For each i , let σ i be such that for every P , σ i ( P ) is P -dominant for i ◮ Let σ ( P ) = � σ 1 ( P 1 ) , ..., σ n ( P n ) � ◮ Let v = g ◦ σ
Proof of theorem Allan Gibbard - Manipulation of voting schemes: a general result (1973) Charlotte Vlek ◮ Fix some strict ordering Q . Let Z ⊆ X Background ◮ For each i , define P i ∗ Z such that for all x , y ∈ X The main result ◮ If x ∈ Z and y ∈ Z then x ( P i ∗ Z ) y iff either zP i y The result for game or both xI i y and xQy forms ◮ If x ∈ Z and y / ∈ Z then x ( P i ∗ Z ) y Proof of theorem ◮ If x / ∈ Z and y / ∈ Z then x ( P i ∗ Z ) y iff xQy Conclusions ◮ Let P ∗ Z = � P 1 ∗ Z , ..., P n ∗ Z � Discussion Literature ◮ define xPy to be x � = y ∧ x = v ( P ∗ { x , y } ) ◮ Show f ( P ) = P is a social welfare function, satisfying all of Arrow’s conditions except non-dicatorship ◮ the dictator for f is a dictator for v = g ◦ σ
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