does individually strategy proof implies coalitionally strategy proof?

Question

Suppose $F$ is a social choice function \begin{equation*}F:N\rightarrow A\end{equation*} where $N=\{1,...,n\}$ is the set of agents and $A$ is a finite set of outcomes.
suppose that $F$ is individually strategy proof, does it imply that $F$ is also coalitionally strategy proof?
edit: I think I can prove it using an argument on $\mbox{Im}(F)$, similar to the Muller-Satterthwaite theorem (for the case $|A|\ge3$). and a preety simple argument for the case $|A|<3$.
This method is not very elegant and spectacular as one might expect from such claim. Is there any other way of proving it?

Answer 1

I am assuming you mean $F: \mathcal{R}^N \rightarrow A$ where $\mathcal{R}$ is the set of preference relations over $A$?

Anyways, the answer is no, in general, strategy-proofness does not imply coalition (or "group") strategy-proofness.

There are some nice "applied" examples in the field of matching where the famous Deferred Acceptance mechanism is known to be strategy-proof (for the proposers) but not coalition strategy-proof. Other nice "applied" examples include public good provision mechanisms such as VCG.

But these examples involve slightly different frameworks (and preference domains) than the one you're interested in. To stick to your abstract social choice framework, the simplest example might be that of a dictatorship that picks outcomes arbitrarily when the dictator is indifferent between a subset of outcomes at the very top of her reported preferences. The dictator does not benefit (stricly) from lying and breaking indifferences at the top (when she is truly indifferent between top outcomes), but doing so may benefit other students, hence benefiting a coalition she may be a part of.

This simple example involves (1) indifferences and (2) a weak notion of coalition improvement where a coalition has an incentive to deviate if it helps some of its members without hurting others.

Examples without (1) and (2) also exist but they are more complicated to describe. (essentially, think of constructing $F$ following kind of "Prisoner's Dilemma pattern" where individuals have a personal incentive to be truthful but can all secure a better outcome by jointly deviating and reporting untruthful preferences).