Enough Information

When we want to measure the amount of information, we believe it is straightforward to do this in terms of bits, a unit that is often used in information theory in which among others the quantification of information is studied. Bits are based on the binary logarithm. For example, if we want to communicate three out of ten different candidates, we need to be able to distinguish between them, so we need at least log(10) bits per candidate and thus 3 log(10) bits to communicate three (see e.g. MacKay, 2003). So, how do we show how much bits of the full profile the manipulator needs to know to never have a disadvantage?

Communication Complexity

Here, we believe that the field of communication complexity is very suitable to look at this problem. Communication complexity tries to quantify how much information is needed for distributed computing. Traditionally, there are two agents, Alice and Bob, who both have their own input string x and y. Now one of them has to compute f(x, y) with the least amount of information exchanged between them. In the context of voting, the function which is computed is the voting rule F and every voteri can be seen as an agent with their input ballot Ri. The communication complexity of F now corresponds to how much bits all

voters in total need to communicate so that every voter has enough information to compute who is winning. The communication can proceed in multiple rounds - formally, this is called

adeterministic protocol, in which all players announce some information to the others voters

based on their input Ri and all bits announced so far. When all players know F(R1, ..., Rn)

the protocol stops. The total minimum amount of bits that need to be sent in the worse case is the communication complexity of F (see e.g. Kushilevitz and Nisan, 1997; Conitzer and Sandholm, 2005).

Conitzer and Sandholm (2005) determined the communication complexity of common voting rules. They showed, for example, that the deterministic communication complexity of the plurality rule is O(nlog(m)), since communicating one (in this case the top) candi- date requires O(log(m)) bits for one player, so if every player announces their top favourite candidate takes O(nlog(m)) bits, and then clearly every player can calculate who wins the election.

By giving an explicit communication protocol, however, we are only able to prove upper bounds of the communication complexity since it is always possible that some more efficient protocol exists. Especially in this context, we are also very interested in the lower bound of the communication complexity of F. Several mathematical tools exist to do this. We will not go into depth into this here, but a nice overview can be found in (Kushilevitz and Nisan, 1997). Conitzer and Sandholm also showed lower bounds of the communication complexity of these voting rules. In particular, they showed that for largem andn, the lower and upper bounds converge for most common voting rules, including the plurality rule, Condorcet, and Borda. The only exception they found was STV.

Connection to STRUCT

Similarly to the upper bound on the communication complexity, we can determine how much bits are needed to communicate the partial profiles R. For example to communicate some

R with ER ∈ TOP(t), we need at most O(n·t·log(m)) bits, since for every voter we need

to communicate t of m alternatives. We can do this for all Σ∈ STRUCT:

Σ Amount of bits needed to communicate

TOPBOT O(nlog(m)) TOP(t) O(ntlog(m)) BOT(t) O(ntlog(m)) COMPL(t) O(tmlog(m))

From the results above and the lower bounds on the communication complexity, we can now formulate the following concrete question:

When the lower bound on the communication complexity is larger than the upper

bound on the number of bits of the profile for some Σ, does there exist some m

and n such that there is some R∈ P(X)n−1 with ER ∈Σ and a true preference

R in which the manipulator has a disadvantage?

The result of this question gives us more knowledge on how the partial information compares to the full information setting. This will tell us more about what kind and extent of the that influence partial information has on manipulation.

We can answer this question in multiple ways. For example, Conitzer and Sandholm showed that the lower bound of Borda is Ω(nmlog(m)) and above we showed that we need at mostO(nlog(m)) bits to communicate some partial profile with a TOPBOT structure. Then we can wonder whether there is some situation in which the manipulator has a disadvantage when only knowing the TOPBOT. As we will show below, this is true for every m ≥5 and n≥4. The proof can be found in the Appendix.

Theorem 6.1. For F being the Borda rule, for any n ≥ 4 and m ≥ 5 there exists some

R0 ∈ L(X)n−1 _{such that G}

TOPBOT(R0) = R and ER ∈ TOPBOT and a trutful profile R

such that the manipulator has a disadvantage in the partial information setting compared to the full information setting.

One way of extending this in future research is to do this for all structures and voting rules. We believe, however, that a more fruitful approach would be to see if the research question above can be proved directly.