Example

Consider a camera and a flash with interacting values to an agent as shown in Table 2. Further, consider that the agent estimates that the camera will sell for $40 with probability 25%, $70 with probability 50%, and $95 with probability 25%. Consider the question of what the agent should bid for the flash (in auction $a_0$). The decision pertaining to the camera would be made via a similar analysis.

Table 2: The table of values for all combination of camera and flash in our example.

	utility
camera alone	$50
flash alone	10
both	100
neither	0

First, the agent samples from the distribution of possible camera prices. When the price of the camera (sold in auction $a_1$) is $70 in the sample:

• $H=(0,0), H_w=(1,0), Y=(\infty,70)$
• $G_w = \mbox{opt}(H_w,Y)$ is the best set of purchases the agent can make with the flash, and assuming the camera costs $70. In this case, the only two options are buying the camera or not. Buying the camera yields a profit of $100 - 70 = 30$. Not buying the camera yields a profit of $10 - 0 = 10$. Thus, $G_w = (0,1)$, and $[v(G_w+H) - G_w \cdot Y] = v(1,1) - (0,1) \cdot (\infty,70) = 100 - 70$.
• Similarly $G = (0,0)$ (since if the flash is not owned, buying the camera yields a profit of $50 - 70 = -20$, and not buying it yields a profit of $0 - 0 = 0$) and $[v(G+H) - G \cdot Y] = 0$.
• $\mbox{val} = 30 - 0 = 30$.

Similarly, when the camera is predicted to cost $40, $\mbox{val} = 60 - 10 = 50$; and when the camera is predicted to cost $95, $\mbox{val} = 10 - 0 = 10$. Thus, we expect that 50% of the camera price samples will suggest a flash value of $30, while 25% will lead to a value of $50 and the other 25% will lead to a value of $10. Thus, the agent will bid $.5\times 30 + .25\times 50 + .25\times 10 = \$30$ for the flash.

Notice that in this analysis of what to bid for the flash, the actual closing price of the flash is irrelevant. The proper bid depends only on the predicted price of the camera. To determine the proper bid for the camera, a similar analysis would be done using the predicted price distribution of the flash.