Suppose you have a majority voting mechanism and a committee composed by a number of players $n$, $C = \{P_1,...,P_n\}$. Each player is required to compute the result of a certain function $f$ and vote accordingly in case the player is honest and vote arbitrary otherwise.
In particular, the players are allowed to vote one of the following options $O = \{HO, DO_1,...,DO_k\}$, where $HO$ represents the only honest option (i.e., the output of $f$) and $DO_1$ to $DO_k$ represent all the possible dishonest options.
In order to incentivize the players to vote honestly I provide a reward $R$ to all the players who agree with the majority.
For example, suppose $C = \{P_1,P_2,P_3\}$, $f = 1$ and $O = \{1,10,100\}$. If $P_1$ and $P_2$ vote 1 and $P_3$ votes 10, both $P_1$ and $P_2$ get the reward $R$.
In general, if no cooperation is assumed among players, $HO$ will receive a number of votes equal to the number of honest players $|C_{honest}|$, while each $DO_i$ for $i = 1,...,k$ is expected to receive (in average) a number of votes equal to the number of dishonest players divided by the number of dishonest options $|C_{dishonest}|/k$ (i.e., the distribution of the votes of the dishonest players among the dishonest options is uniform).
So, as far as $|C_{honest}| > |C_{dishonest}|/k$ voting honestly is the rational strategy for each player.
Is this argument correct?
Instead, if a player does not know if $|C_{honest}| > |C_{dishonest}|/k$ is satisfied, then the only possible consideration, under the no cooperation assumption, is that each option is expected to receive (in average) a number of votes equal to $|C|/(k+1)$ (i.e., the distribution of the votes of the players among the options is uniform). In this case there is no rational strategy and any strategy is equivalent. Is this correct?
Another "degenerate" scenario I see is when $k = 1$ (i.e., only one dishonest option). In this case, having $|C_{honest}| > |C_{dishonest}|$ is required to guarantee that voting honesty is the rational strategy for each player. However, if I assume that players are composed by an honest majority, then there is no need to create an incentive to have an honest majority (since it is already an assumption) so as to have the voting mechanism selecting $HO$ as a winner.
It seems to me that when $k = 1$ the described voting and incentive mechanisms are useless. Is this correct?
If so, how can I design a voting and an incentive mechanism that implies the majority of players being honest (and does not require it as an assumption) when $k = 1$? Is it possible?