What strategy should Alice choose to win this game? The following game was proposed to me:


*

*Alice has a coin which has a probability $p$ of landing on heads, $1-p$ of landing on tails. Alice doesn't know $p$. Alice is not allowed to lie.

*Bob knows $p$. Bob is allowed to lie.

*Alice tells Bob two functions $f, g: [0,1]\mapsto\mathbb R$ and asks Bob for the value of $p.$ Let Bob's answer be $\pi$.

*Alice throws the coin. If it lands on heads, Bob is awarded $f(\pi)$ points, if it lands on tails, Bob is awarded $g(\pi)$ points.


Alice wants to find a pair of functions $f,g$ such that if Bob tries to maximize the expected value for the number of points he is awarded, then $\pi=p$.
Is there a simple way to describe the entire solution space?

This resembles mechanism design problems in economics. You, mechanism designer, are trying to construct a mechanism, map from answers to outcomes and transfers, such that agents participating in the mechanism have no incentive to lie about their private information.
In your context, suppose $f,g$ are already set. Then Bob, when choosing $\pi$ to answer, solves $\max_{\pi\in[0,1]}pf(\pi)+(1-p)g(\pi)$ (notice Bob knows $p$). Assume first-order conditions describe the solution to this problem. Then Bob answers $\pi$ such that $pf'(\pi)+(1-p)g'(\pi)=0$. Since your (Alice's) goal is to induce truth-telling, Alice wants to set $f$ and $g$ such that $pf'(p)+(1-p)g'(p)=0$ and since Alice does not know $p$, she wants to set $f$ and $g$ such that $pf'(p)+(1-p)g'(p)=0$ for all $p\in[0,1]$ (with the continuity, differentiability, and concavity assumptions made on the way here).
If you do not like the assumptions I made throughout since you want to describe the entire solution space, then you simply write that $f$ and $g$ constitute your solution if $p\in\arg\max_{\pi\in[0,1]}pf(\pi)+(1-p)g(\pi)$ for all $p\in[0,1]$.
