Signalling Game, Multiple Signals

Question

I am solving a three-stage game in a supply chain with one buyer and one supplier. The supplier has private information on its production capacity. The supplier also has the option to sell to the market directly. As of now, the game sequences are as follows. In the first stage, the supplier decides on the wholesale price, then the buyer takes the wholesale price as a signal, updates its belief about the production capacity of the supplier and order some of that capacity. In the last stage, the buyer decides, how much it wants to sell to the market directly, and the supplier decides how much to sell to the market if any capacity remains. For now, the only signal is the wholesale price. I want to add a second signal to the first stage. In other words, the supplier not only sets the wholesale price but also puts a limit on the buyer's order quantity. In the next stage, the buyer solves its expected utility by considering two signals.

Has someone heard about a similar story with 2 signals at one stage? Thanks

Answer 1

From the perspective of the buyer, he is receiving 1 two-dimensional signal. After observing a combination of the wholesale price and limit order, the buyer can update their beliefs about the supplier's capacity using Bayes rule. Let me show it:

Let $c\in [0,1]$ be the supplier's capacity (just for simplicity of notation I assumed it is in the interval from 0 to 1) and assume $c$ is distributed according to $\mu_0(c)$, thus $\mu_0$ is prior belief of the buyer that the capacity is $c$. Then each type of supplier will optimally choose a signal $(p_w(c), \bar o(c))$ of the wholesale price and the limit on buyer's order. These are functions from the production capacity $c$ (or more generally, the suppler's private information) into $\mathbb{R}^2$ (if some types of suppliers use mixed strategies, then $(p_w(c), \bar o(c))$ are functions that return distributions supported in $\mathbb{R}^2$.

After receiving a signal say $(p_w, \bar o)$ the buyer's posterior belief, denoted $\mu_1(c)$, is given by:

$$\mu_1(c|(p_w, \bar o))=\frac{Prob((p_w(c), \bar o(c))=(p_w, \bar o))\mu_0(c)}{\int_{[0,1]} Prob((p_w(t), \bar o(t))=(p_w, \bar o))\mu_0(t)dt}$$

That is the ratio of the probability of receiving signal $(p_w, \bar o)$ from a supplier with capacity $c$ over the total probability of receiving the same signal; from any type of supplier. Note that if the supplier with capacity $c$ is using a pure strategy, then $Prob((p_w(c), \bar o(c))=(p_w, \bar o)$ is either $0$ or $1$.

Given the 2 dimensional signal, I would suspect that one of the following 3 scenarios will occur:

1. The buyer learns the capacity form the signal (for example if the function $f(c)=(p_w(c), \bar o(c))$ is injective from $\mathbb{R}$ into $\mathbb{R}^2$. (notice that this can occur even if one or both of the functions $p_w(c), \bar o(c)$ are not injective).

2. The function $\bar o(c)$ is constant, so you revert back to the case with a 1-D signal since the buyer learns nothing from $\bar o(c)$.

3. Suppliers choose mixed strategies. In that case, finding $(p_w(c), \bar o(c))$ will be a challenge because the set of distributions supported in $\mathbb{R}^2$ is quite large.

Good luck!