Asymptotic Strategy proofness of the Plurality and the Run o Rules

In this paper we prove that the plurality rule and the run o procedure are asymptotically strategy proof for any number of alternatives and that the ratio of the number of all manipulable pro les upon the to tal number of pro les in both cases is it the order of O p n Introduction The well known impossibility theorem of Gibbard and Satterthwaite states that every non dictatorial social choice function is manipulable This result is also valid for social choice rules i e for those social choice func tions for which the choice set is not necessary singleton valued see and the literature there Therefore we have to accept a certain degree of ma nipulability while trying to minimize it as much as possible In this respect the social choice rules for which the probability of possibility to manipulate tends to zero as the number of agents grows are especially attractive and must be used each time when the number of agents is large We call such rules asymptotically non manipulable or asymptotically strategy proof Since all non dictatorial social choice rules are manipulable it would be useful to know which social choice rules are manipulable to a lesser extent To this end an experimental approach to the study of the degree of manip ulability of social choice rules was initiated in and continued in where di erent rules were investigated by means of computer experiments An other approach for evaluating the degree of manipulability is theoretical The author in proved the asymptotic strategy proofness of the plurality rule m alternatives and the run o procedure m alternatives and proved the asymptotic strategy proofness of the majoritarian compromise In all cases the asymptotics for the ratio of the number of manipulable pro les upon the total number of pro les appeared to be O p n These results together with the results of the aforementioned experi ments give hope that all natural rules are asymptotically strategy proof where the appropriate content for the concept of being natural is yet to be de ned The same speed of convergence to zero of the probability to obtain a manipulable pro le for all these rules is a rather convincing evidence that there must be a common reason for them to be asymptotically strategy proof Our results in this paper also support this vaguely stated conjecture The concept of asymptotic strategy proofness stipulates that as the num ber of agents goes to in nity then with probability approaching no one agent can gain an advantage by misrepresenting her true preferences The stronger condition that we are going to introduce stipulates that as the number of agents goes to in nity then with probability approaching a pro le chosen at random will be stable in the sense that no one agent can change the result of elections at all neither to her advantage nor to her disadvantage We call such rules asymptotically fool proof A fool proof so cial choice rule is not a ected by a random mistake that an agent might do casting her ballot In this paper we prove that the plurality rule and the run o procedure are asymptotically fool proof for any number of alternatives and that the ratio of the number of all manipulable pro les upon the total number of pro les in both cases is in the order of O p n For the plurality this asymptotics is exact It should be noted that manipulability by coalitions of voters is a separate problem see for example and we do not touch it in this paper Basic Concepts Let IN stand for the set of all positive integers and let Im f mg The elements of Im are called alternatives Let N be a nite society of n agents confronted with the choice among these alternatives By L Im we denote the set of all linear orders on Im they represent the preferences of agents over Im If Ri L Im represents the preferences of the i th agent then by aRib where a b Im we denote that this agent prefers a to b The elements of the cartesian product L Im n L Im L Im n times are called n pro les or simply pro les They represent the collection of pref erences of all agents in N A family of correspondences F fFng n IN Fn L Im n P Im where P Im is the power set of Im we will call a social choice rule Normally it is assumed that F represents a certain algorithm which on accepting a positive integer n and an n pro le R L Im n outputs a subset Fn R of Im In this paper rst we deal with the plurality rule which in our under standing on accepting n IN and an n pro le R outputs the set of those alternatives which received a maximal number of agents rst preferences The number of rst preferences that an alternative a Im received will be called the score of this alternative and will be denoted by score a Thus in this terminology we may say that the plurality winners are the alternatives with the maximal score De nition Let R R Rn be a pro le We say that the pro le R is manipulable if there exists a linear order R i such that for a pro le R R R i Rn where R i replaces Ri we have one of the two possibilities For some a Fn R it is true that aRib for all b Fn R The best element of F R relative to Ri is the same as the best element of Fn R relative to Ri but Fn R is strictly contained in Fn R The rational behind such de nition is as follows Suppose that the alter natives chosen if we have more than one of them will further participate in a lottery with equal chances to win Then every outcome can be viewed as a probability vector Introducing lexicographic ordering on probability vectors we assume that an agent will prefer one outcome to another if the probability vector of the rst outcome is lexicographically earlier than that of the second This de nition was implicitely suggested in De nition Let R R Rn be a pro le We say that the pro le R is unstable if there exists a linear order R i such that for a pro le R R R i Rn where R i replaces Ri we have Fn R Fn R Clearly every manipulable pro le is unstable but the reverse is not always true The Kelly s index of manipulability of F as suggested in is KF n m dF n m m n where dF n m is the total number of all manipulable pro les Let us also de ne the index of instability of F by the formula LF n m eF n m m n where eF n m is the total number of all unstable pro les If we choose pro les at random from the uniform distribution then of course KF n m and LF n m have meanings of the probabilities to obtain a manipulable or unstable pro le respectively De nition We say that a social choice rule F is asymptotically strategy proof for m alternatives if KF n m as n and asymptotically fool proof for m alternatives if LF n m as n Of course KF n m LF n m and asymptotic fool proofness implies asymptotic strategy proofness The two concepts as we will show later coincide for the plurality rule for m alternatives But for the plurality it is much easier to count unstable pro les that is why this concept is so important The Plurality Rule Manipulable Versus Unstable Pro les We say that a pro le R L Im n has the type k km where k k km if there is a permutation a am of the alternatives from Im such that the score of ai is ki Proposition A pro le is unstable for the plurality rule if and only if it is of the type k km where k k Proof If k k i e the alternative a has the score which is greater by at least than the score of the alternative a then no one agent can make a a winner or make the scores of a and a equal On the other hand if the di erence in scores of a and a is less than or equal to the result of these elections may be changed by any agent who had a as her rst preference We will refer to the pro les speci ed in Proposition as to the pro les of class A when k k to the pro les of class B when k k and to the pro les of class C when k k Proposition A pro le of the type k km of class A is always manip ulable for the plurality rule unless for some s m we have k ks n s Proof Clearly the condition k ks n s in case s m implies ks km It is easy to check that the pro les satisfying the above condition are not manipulable Suppose that for a pro le R this condition does not hold Then for some s m we have k ks ks Then the alternatives a as get equal score which is maximal Thus the set of alternatives fa asg will be chosen Let us consider an agent whose rst preference is as Suppose that a is her best alternative among the chosen ones Then she can manipulate by pretending that a is her rst preference swapping it with as This will make a the sole winner to her advantage Proposition Let R be a pro le of the type k km of class B and k k ks ks Suppose that the score of the alternative ai is ki Then R is manipulable for the plurality rule if and only if there exist two di erent numbers i and s j such ki and ajRia Proof The agents whose rst preference is a cannot manipulate An agent whose rst preference is ai i is in a position to manipulate if ajRia for some j i such that j s She might declare that her top alternative is aj instead of ai in which case it will also be chosen together with a This is to her advantage Proposition Let R be a pro le of the type k km of class B Then it is manipulable for the plurality rule with probability not smaller then m t where t k km Proof Assuming k k ks ks suppose that nobody can manipulate in such a way that a is selected together with a Then the pro le must satisfy the following property each agent whose rst preference is di erent from a and a prefers a to a This occurs with probability t where t k km The probability that nobody can manipulate in such a way that aj is selected for j s is not greater than t for each j Since s m the proposition is proved Proposition A pro le of class C is never manipulable Proof If the di erence in scores of a and a is exactly then the only way in which the result may be changed is that an agent who voted for a makes the scores of a and a equal by placing a rst Clearly this is not to her advantage and hence it is not a manipulation Lemma There exists a positive real number such that the proba bility for a random pro le R to be of the type k km with km n exponentially decreases as n Proof The number of all possible allocations of rst preferences is m The number of all allocations of rst preferences in such a way that at least one alternative receives less than or equal to s n rst preferences can be estimated as being less than or equal to m m n n m n n s m n s Thus the probability to get such a pro le is bounded above by the number