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Let $A$ be a finite set of alternatives with $|A|\ge 3$, and let each agent report a strict total order on $A$. A profile is the tuple of these orders. The domain is unrestricted: every tuple of orders is allowed. A deterministic social choice function $f$ selects one alternative from each profile. It is onto if every alternative is selected at some profile. It is strategyproof if no agent, holding all other reports fixed, can obtain an outcome strictly preferred according to its true order by reporting another order. A dictatorship in social choice means that one fixed agent's highest-ranked alternative is always selected, regardless of all other reports.

The Gibbard-Satterthwaite theorem says that every onto, deterministic strategyproof social choice function on this unrestricted domain is a dictatorship in social choice. Equivalently, with at least three possible alternatives an onto nondictatorial rule must permit manipulation. Both the number of alternatives and the unrestricted preference domain are essential hypotheses.

For the requested top-bottom decisiveness lemma, suppose $f(R,S)=a$, with $a$ top in $R$ and bottom in $S$. Fix any $R^{\prime }$ with $a$ top. If $f(R^{\prime },S)\ne a$, agent $1$ with true order $R^{\prime }$ could report $R$ and obtain its best outcome $a$, contrary to strategyproofness. Hence $f(R^{\prime },S)=a$. Now fix any $S^{\prime }$. If $f(R^{\prime },S^{\prime })=b\ne a$, agent $2$ with true order $S$ could report $S^{\prime }$ and obtain $b$, which is better than its bottom alternative $a$. ThereforeCall this agent $1$ being decisive for $a$. The same argument with agents interchanged applies to agent $2$.

Here is a complete proof of the two-agent theorem. First, onto plus strategyproofness gives unanimity. If $a$ occurs at some profile, change agent $1$ to any order with $a$ top: a different outcome would allow that agent to obtain $a$ by returning to its old report. Then change agent $2$ to any order with $a$ top in the same way. Thus every profile with both agents ranking $a$ top selects $a$.

We also need Pareto efficiency. A useful rank-raising monotonicity lemma follows from strategyproofness: if the current outcome is $b$, and one agent changes its order without placing above $b$ any alternative previously below $b$, the outcome remains $b$. Otherwise an outcome $c\ne b$ would satisfy $b\succ c$ in the old order, since a profitable deviation is forbidden there, but $c{\succ }^{\prime }b$ in the new order, since reporting the old order is forbidden there. Those comparisons contradict the allowed change. If both agents prefer $a$ to a selected $b$, change their orders one at a time to put $a$ first and $b$ second. These changes only raise $b$ relative to other alternatives, so the outcome remains $b$, contradicting unanimity for $a$. Therefore an alternative unanimously dominated by another cannot be selected.

Choose distinct $a,b$. Give agent $1$ order $b\succ a\succ \text{the rest}$ and agent $2$ order $a\succ b\succ \text{the rest}$. Pareto efficiency forces the outcome to be $a$ or $b$. If it is $b$, move $b$ to the bottom of agent $2$'s order while keeping $a$ top. A switch to $a$ would be a profitable deviation for its old true order. Every other alternative is unanimously dominated by $a$. Hence the outcome stays $b$, and the top-bottom decisiveness lemma makes agent $1$ decisive for $b$. If the outcome is $a$, the symmetric argument makes agent $2$ decisive for $a$. Consequently some agent is decisive for some alternative. Rename that agent $1$ and that alternative $a$.

Both agents cannot be decisive for this same $a$. To see this, choose $b,c\ne a$ distinct, with every remaining alternative below these three, and considerAt $(R,S^{\prime })$, agent $1$ can obtain $a$ by placing it top, so strategyproofness restricts the outcome to $a$ or $b$. Both agents prefer $b$ to $a$, so Pareto efficiency forces $b$. At $(R^{\prime },S^{\prime })$, agent $1$ still ranks $b$ top and can obtain it by reporting $R$, so the outcome must again be $b$. On the other hand, at $(R^{\prime },S)$, decisiveness of agent $2$ for $a$ restricts the outcome to $a$ or $c$. Both agents prefer $c$ to $a$, so it must be $c$. Since $c$ is top in $S^{\prime }$, agent $2$ can obtain it at $(R^{\prime },S^{\prime })$ by reporting $S$, forcing outcome $c$ there. This contradiction proves that agent $2$ is not decisive for $a$.

Now fix $b\ne a$ and any agent-$1$ order $R$ with $b$ first and $a$ second. At every $(R,S)$, agent $1$ can force $a$ by placing it top, so the outcome must lie in $\{a,b\}$. If $a$ occurs for even one report $S$, changing agent $2$ to an order with $a$ first and $b$ second preserves $a$, because it can return to $S$ to obtain its top choice. Lower $a$ to the bottom of agent $1$'s order, keeping $b$ top. A switch to $b$ would be a profitable deviation under its old order, and any other outcome is unanimously dominated by $b$. Thus $a$ remains selected at a profile where agent $2$ ranks it top and agent $1$ ranks it bottom. The top-bottom decisiveness lemma would make agent $2$ decisive for $a$, a contradiction.

Therefore every $(R,S)$ with agent $1$ ranking $b$ first and $a$ second selects $b$. Choose $S$ with $b$ bottom and apply the top-bottom decisiveness lemma once more: agent $1$ is decisive for $b$ in every order placing $b$ top. This holds for each $b\ne a$, and it already holds for $a$. Agent $1$ is a dictator in social choice, proving the two-agent theorem.

With three agents and only two alternatives, the two-alternative majority rule is onto and strategyproof and has no dictator in social choice. An agent whose vote is not pivotal cannot change the outcome. A pivotal agent obtains its preferred alternative by voting truthfully and its less preferred alternative by reversing its vote. No one is a dictator in social choice, since the other two agents can both oppose its preference. There is no tie with three agents. This does not contradict the theorem because its hypothesis $|A|\ge 3$ fails.

With an odd number of agents and two alternatives, select the alternative ranked first by a strict majority. The rule is onto, strategyproof, and for at least three agents has no dictator in social choice. A pivotal agent already gets its preferred alternative by reporting truthfully; a nonpivotal agent cannot change the outcome alone.