# Bayes Nash Equilibrium in a game with continuous actions

I am attempting to think through a particular type of game with continuous strategies, with Bayes Nash equilibrium as the solution concept. I first describe the game below, followed by questions.

## Game

Consider the following game between two agents: A and B.

• Agent A draws a cost $k$ from a distribution $F(k)$, which has support on $[0,\overline{k}]$. Knowing this cost, she chooses an action $x\in [0,\infty]$ and thereby incurs a cost $k\cdot x$.

• The choice of $x$ produces a benefit $R(x)$ for agent B, with $R(0)=0$, $R^\prime(x)>0$ and $R^{\prime\prime}(x)<0$.

• $R(x)$ is not observed until the game is completed. Instead, A is paid by B for her effort based on a noisy but strictly informative signal $\xi(x)$. Higher $\xi$ always indicates higher $x$ on average.

• Specifically, B has a prior distribution $\pi (x)$ over $x$ and observes $\xi(x)$. B always pays A her expected product $E(R(x)|\pi,\xi)$ so that B expects a payoff of zero for any value of the signal.

• Finally, in Bayes Nash equilibrium, we require that $\pi (x)$ is "correct": i.e., given the distribution of costs $F(k)$ and the expected payments as a function of $x$ and $\pi (x)$, actions $x$ actually do have the distribution $\pi(x)$.

## Questions

1. There appears to always be an equilibrium to this game with a degenerate prior $\pi(x)$ that has all its mass on zero. In this case, $E(R(x)|\pi,\xi)=0$ for all $x$, and A would never choose $x>0$, confirming the prior. Do you agree?

2. What are some non-trivial sufficient conditions for there to be additional equilibria in pure strategies (i.e., with $x>0$ in some states of the world)?

3. How would one characterize these other equilibria if they exist? For example, how would the set of equilibria change if the benefit function increased from $R(x)$ to $\hat{R}(x)$ = $c\cdot R(x)$ where c is a constant greater than one?

• I am aware of examples of this kind of game with discrete actions $x$. These are comparatively easy to analyze. What I am struggling with here is how to think about equilibria with continuous actions and therefore also a continuous prior $\pi(x)$. The fixed point problem seems much trickier in this case.