Consider a problem that looks for a $x$ that can make the following problem into some equilibrium state (similar to an equilibrium solution to a min-max problem in game theory)
$$ \max_x f(x)$$ $$ \min_x g(x)$$
subject to $$ f(x) \geq a$$ $$ g(x) \leq b $$
where $f$ and $g$ are real-valued functions, and $x$ is from some set.
For example, as a customer, I am trying to choose a product of a given type to buy, but have to balance between product quality and price. I will make the choice by making choices for the factors that the quality and/or price of the product depend on.
- $f$ represents the quality of a product, and $a$ represents the lowest quality acceptable,
- $g$ represents the price of the product, and $b$ represents the available budget.
- $x$ is multi-dimensional and consists of factors that the quality and/or price of the product depend on, for example, the product model (thus manufacturer), seller (base price, purchase charge, shipping fee, tax, coupon, promo code, and any other discount), payment method (cashback credit cards, gift cards, and which card if i have multiple), and purchase timing.
The ideal solution is to find some $x$ so that $f(x)$ and $g(x)$ are in equilibrium in some sense similar to the solution to a minmax problem in game theory, instead of maximizing $f$ and minimizing $g$ separately.
The constraints are not superficial. When there is an $x$ so that $f(x)$ and $g(x)$ are in equilibrium, $f(x)$ and $g(x)$ may not satisfy the constraints, and there should be a compromise in what we mean by "equilibrium" by constraints. Note: If the constraints make the problem complicated, you may temporarily ignore them. I think we should start with a simpler problem, and understand them before considering a more complicated one.
My questions:
Is the problem a game theory problem or optimization problem?
Have there been well-defined/formulated problems similar to my problem?
The problems I have heard in game theory is the min-max (or max-min) problem of the form $\min_x \max_y h(x,y)$, and the above problem doesn't seem to fall into the min-max problem form, because my problem has two objective functions (I am still in the process of understanding my problem and I am not sure if it has been well-formulated)
In the optimization problems that I have heard, there is only one objective function which is either minimized or maximized over some independent variables.
What are some methods to solve such a problem in general?
Thanks.
From what you say it sounds like it should be a problem of constrained optimization. as Shane has said, you need to take a stand on the utility function, how the consumer balances benefits and costs. The standard way in economics is to assume there is a price (usually linear, but it doesn't have to be) http://www.unc.edu/~normanp/unc410week3.pdf Then your constraint set is your income, and you maximize your utility (like your f function) subject to the prices of the goods and your budget set (the set x belongs to). Remember that the minimax result of game theory is for zero-sum games, that is, what ever you win I lose, so implies that your f and g functions are identical, so even if there is a game-theoretic interpretation, it's not clear to me that minimax would be the correct solution anyway. Way more details in David Krep's book A Course in Microecnomic Theory.