Static game with complete and incomplete information

Question

I am currently trying to learn game theory on my own.

I have a question regarding the solution methods for static games with complete information vs that of incomplete information. The textbook which I have for my reference often associates static games with incomplete information with a probability belief. However, right now, I am thinking of a game whereby a player does not have any idea about the other player's payoff function.

In this case, how can I solve the problem? Would the approach using the first order equations work here?

Answer 1

Well, it gets complicated. There are simple cases, of course, where you have a dominant strategy where you don't care what the other person does, but that's not very interesting. Basically, you need some idea about the other person's payoff function. Approaches to knowing very little, that is, not knowing the probability distribution of the other person's payoffs are associated with the terms Knightian Uncertainty, the Ellsberg Paradox, and Gilboa-Schmeidly preferences. For an example of a paper that tries to put these into game theory, see http://www.bus.indiana.edu/BEPP/documents/FrankRiedel.pdf, but it is rather advanced.

Answer 2

Sure. Think of an auction as an example that unifies both of your strands of thought. In an auction, each agent's expected profit function is:

$$\pi_i=\mathbb{P}(b_i > b_j : \forall j\neq i)\cdot (v_i -b_i) $$

where $b_i$ is agent $i$'s bid, and $v_i$ is agent $i$'s valuation of the good being sold. In other words, it is simply the probability their bid is the highest, times the margin they'd get (their value less their bid) conditional upon winning.

In this case, it is customary to have some prior belief over the distribution that opponents' valuations $v_{-i}$ are drawn from (in the simplest case it is assumed all our valuations are drawn from the same distribution $F(\cdot)$. Since bids turn out to be increasing functions of valuation, a prior over their valuation gives me also a unique prior over their choice of strategy (bid).

Yet this is also equivalent to uncertainty over their utility function! If I don't know your valuation $v_{j}$, I don't know your expected utility function $\pi_j$ as given above. In this case, the two notions are intimately related.

Now, where things differ is if I don't know my own payoffs. That's where things can get a little more hairy. In the case you're working on though, I'd suggest just diving into into the FOCs. That approach almost never hurts.