I recently read an article on Management Science, in which supermodular functions were used. So what are supermodular functions? I heard that there is a low-module function (some people call it a sub-module function). What is it?
We have to solve them one by one, otherwise we will not understand the paper later.
1. Basic meaning
The low-module function is used to describe substitutable products. An increase in the sales of one product causes a decrease in the sales of another product.
2. Theoretical background
Basic introduction: The supermodel game is closely related to substitution and complementarity, and its setting depends on the lattice structure (lattice is generally called the lattice). This theory was first proposed by operations researcher Topkins (1978, 1979, 1998). In economics, it was Bulow et al. (1985) who introduced strategic complementarity and strategic substitution. Nobel Prize winners Milgrom and Roberts (1990) and Vives (1990) of the same period further integrated their achievements. Shapley studied content related to cooperative games in many articles in 1995, 1962, 1971, and 1972, which included the idea of complementary substitution in supermodel games.
3. Relevant knowledge
First of all, we need to recognize this symbol that looks like a greater than or equal to ( partial order ). This symbol can be temporarily understood as a greater than or equal to sign. It has reflexivity, antisymmetry and transitivity . However, if you take any two elements, there may be no way to compare them ( For example, I am not sure about my preferences for certain products).
What is lattice? For all x, y in X, there exists:
- Supremum bound : z (in lattice) is larger than x and y, but it is the smallest among those larger than them.
- Lower bound : z (in lattice) is smaller than x and y, but it is the largest among them.
In binary Euclidean space, any two points (x, y) satisfy partial order, but the two points cannot be compared (for example, one is bigger in x and the other is bigger in y). If X is a circle ( in the coordinate system ), it is not a lattice, because z is not in the lattice. If X is a square, it is a lattice.
Complete lattice (complete lattice): Any number of elements have a supremum and an indefinite bound.
The supermodel game is also divided into non-cooperative supermodel game and cooperative supermodel game. Both of these are based on the lattices theory in mathematical description , so they are closely related. Both Cournot competition and Bertrand competition can be studied using the supermodel game.
S^N is a Cartesian product (low dimensions eventually form high dimensions). When everyone's strategy set is multiplied, it ends up being n-dimensional (x1, x2,..., xn).
The mathematical symbol like v means taking the maximum in each dimension; the one like the inverted v means taking the minimum in each dimension.
Interpretation: players i and j are pairwise strategic complements : for i, all si are strictly greater than ti, for j, sj is also strictly greater than tj , and people other than i and j decide a strategy. Then satisfy equations (1) and (2).
(1) The left side of the formula: the difference in utility between i’s choice of strategies si and ti (other people’s choices remain unchanged, including j’s choice of sj). This can be understood as the marginal contribution of the difference between si’s strategy and ti’s strategy to utility ui, because the previous It is said that si is strictly greater than ti. The equation on the left is greater than the change in i's utility when j chooses tj on the right. (2) is the same, the protagonist changes from i to j.
The intuitive interpretation is that when j chooses sj instead of tj, the marginal utility of i will be greater. It can also be understood that when j changes his strategy from small tj to large sj, the incentive for i to improve itself becomes greater (from ti to si). The meaning here is: the directions in which people i and j adjust their incentives are consistent. With directionality and monotonicity , it is helpful to solve the equilibrium.
If it is a strategic substitution game, the two symbols are changed to the opposite.
For a game of strategic complementarity to exist, the above two assumptions need to be met:
Assumption 1 can ensure that everyone's maximum can be obtained. Assumption 2 is true when the grid is linear .
4. Supermodular function
5. Properties of supermodular functions
The picture on the right is a two-dimensional space with L=R^2 . The interpretation of the definition in the picture is that the value of the red point and the value of the blue point are greater than the sum of the values of the two yellow points.
understand:
1. The cross-derivative of the supermodular function is greater than or equal to 0 (easy to judge) (this can help us determine whether it is a supermodular function). In addition, any one-variable function is supermodular, and the formula always holds. Even if it is multivariate, it is easy to judge cross-leads.
2. The supermodular function can be discrete (widely used) , the domain can be discontinuous, and the scope of application is wider.
3. The supermodular function already contains pairwise strategic complements (strong conditions) between any two people (as mentioned earlier) .
Supermodular function properties :
1. Pure strategy Nash equilibrium exists.
2. Not only does it exist, but there is also a lattice structure (ie pure strategy NE has a maximum equilibrium and a minimum equilibrium). Dynamic perspective : If the starting point is appropriate, it will converge to the maximum/minimum equilibrium. There are some parameters that can help with comparative static analysis . , can interpret the economic meaning.
Prove the existence of functions (Tarski’s fixed point theorem )
NE is equivalent to the fixed point of some optimal reactions. If we have some properties, we can guarantee that the fixed point exists (that is, NE exists). This theorem talks about the characteristics of monotonicity (as shown on the left): x is a complete lattice, f is a monotonically increasing mapping (order-preserving transformation) , there must be a fixed point (the intersection with the 45° line), and there is no The moving points together form a complete lattice.
The converse is not true (as shown in the picture on the right): it is a monotonically decreasing mapping, which does not necessarily have a fixed point. It can "jump down" directly and does not intersect with the 45° line .
6. Application examples
6.1 Bertrand competition ( price competition )
p_i is a complete lattice; Q is the demand; it is a strategic complementary game (core condition: it can be known from the cross derivative = wij>0). If the price of one manufacturer decreases, the other manufacturer also has incentives to decrease. Other bits and pieces of conditions are also easy to verify.
The same applies to Cournot competition, but the cross-derivative is less than 0, which is a strategic substitution game: for example, one manufacturer's output quantity is adjusted upward, and the output quantity of another manufacturer is adjusted downward (it can be converted into a strategic complementarity game by adjusting the sign and distorting the lattice).
6.2 Diamond’s Search Model
a_i is on the complete lattice; g(0)=0, g(1)=1; the optimal response is unique and monotonically increasing, indicating that the strategies are complementary. θ and ai have a complementary relationship, which can be interpreted as the higher the return, the higher the level of everyone's efforts.
Reprinted in: How to simply explain the "Supermodular Game"? - Know almost