How To Solve Game Theory Problems With Fmincon In Matlab

To solve game theory problems with fmincon in MATLAB, formulate your problem as an optimization, define an objective function representing the payoffs and constraints related to game’s rules, then use fmincon to find the optimal strategies.

Ever wondered how to apply mathematical optimization to strategic scenarios? We often encounter situations that can be modeled using game theory, where multiple players make choices influencing each other’s outcomes. This blog post explores how to solve game theory problems with fmincon in MATLAB.

Fmincon, a powerful optimization tool within MATLAB, allows us to find optimal solutions for constrained problems. Translating game theory problems into an optimization format opens up possibilities for simulating and analyzing complex strategic situations.

How to Solve Game Theory Problems with fmincon in MATLAB

Let’s dive into the exciting world of game theory! Imagine you’re playing a game, but instead of just having fun, you want to figure out the best strategies to win. That’s where game theory comes in. It’s a way to mathematically analyze situations where multiple people or groups make decisions, and their choices affect each other. Now, to solve these game theory problems, we can use a powerful tool in MATLAB called fmincon. Think of fmincon as a super-smart puzzle solver that can find the best answers to complex problems, especially when those answers involve constraints or limits.

Understanding Game Theory Basics

Before we jump into MATLAB, let’s make sure we understand some key game theory ideas. Games in game theory aren’t just about fun; they are about strategic interactions. Here are some essential elements:

Players

These are the people or groups making decisions. Each player’s choice influences the outcome for everyone.

Strategies

These are the different actions a player can take. A player may have multiple options to choose from.

Payoffs

This represents what each player gets as a result of everyone’s choices. Payoffs can be things like money, points, or anything else you can assign a value to.

Nash Equilibrium

This is the main goal in many game theory situations. It’s a state where no player can improve their own payoff by changing their strategy alone, assuming everyone else keeps their strategies the same. It’s like a stable point in the game.

Let’s consider a simple example: The Prisoner’s Dilemma. Two suspects are arrested for a crime. They are held separately and each has two choices: cooperate with the other person or defect and betray the other person. Here’s how the outcomes can look:

If both cooperate, they get a moderate punishment (say 3 years).
If one cooperates and the other defects, the defector goes free and the cooperator gets a harsh punishment (say 10 years).
If both defect, they both get a moderate-heavy punishment (say 5 years).

The Nash Equilibrium here is for both prisoners to defect, even though they would both be better off cooperating. This highlights an important aspect of game theory, where individual rational choices can lead to outcomes that are not the best for everyone involved.

What is fmincon?

So, where does MATLAB’s fmincon fit in? fmincon stands for “constrained minimization function.” It’s a tool designed to find the minimum value of a function, but it also allows you to specify limits or conditions (constraints) that the solution must satisfy. In game theory, we can often frame finding a Nash equilibrium as an optimization problem, where we want to minimize some function related to the players’ choices. These choices also have to meet some constraints like, probabilities must add up to one. This is where the constrained optimization of fmincon comes to help. Let’s break this down a bit further:

Minimization

Our goal is to find the smallest value of an objective function. This function represents the ‘cost’ or the ‘error’ we want to make as small as possible. In game theory, it is related to the payoff, or change of payoff of one or more players.

Constrained

We have rules or limitations that the solution must follow. These are constraints. For instance, if we’re dealing with probabilities, these need to be between 0 and 1, and must add up to 1.

fmincon can handle different types of constraints:

Linear inequalities: Equations like Ax <= b, where A is a matrix, x is the variables, and b is another set of values.
Linear equalities: Equations like Aeqx = beq, where Aeq is a matrix, x is the variables and beq is a set of values.
Nonlinear inequalities: More complex inequality relationships involving functions of the variables.
Nonlinear equalities: More complex equality relationships involving functions of the variables.
Bounds: Simple limits to the variables, such as x being greater than or equal to zero and less than or equal to 1.

By using fmincon, we can find the best strategies that players can use within the set constraints.

Steps to Solving Game Theory Problems with fmincon

Alright, let’s get practical! Here’s how you’d typically go about using fmincon to solve a game theory problem:

1. Define the Game

First, clearly state your game. What are the players? What are their possible strategies? What are their payoffs for each possible outcome? Write out the payoff matrix or the function relating each player’s decision to the payoffs. For an example, Let’s consider a 2×2 game between player A and player B. Player A can choose between strategy 1 and 2. Same goes for player B. Let’s imagine the payoff for player A is given by the matrix:

A = [3 0
1 2]

This matrix means if player A uses strategy 1, and player B chooses strategy 1, player A receives a payoff of 3. If player A uses strategy 1 and player B chooses strategy 2, player A receives a payoff of 0. If player A uses strategy 2 and player B chooses strategy 1, player A receives a payoff of 1. And if player A chooses strategy 2 and player B chooses strategy 2, player A receives a payoff of 2.

2. Formulate as an Optimization Problem

Translate the game into something fmincon can work with. This involves two key things:

Objective Function: You’ll need to define a function that you want to minimize. This function should be designed in such a way that when you find its minimum, it gives you a Nash equilibrium. This is not always straightforward, and depends on the game type and what information we wish to analyze.
Constraints: Define the limitations of the problem. These often involve things like probability constraints where sum of the probabilities of each strategy must add to 1. For example, the probability that a player chooses strategy 1 plus the probability they choose strategy 2 must equal 1.

3. Write the MATLAB Code

Here’s where the magic happens! You’ll need to code this optimization problem in MATLAB using fmincon. Here’s an example, based on the payoff matrix example above:

    
function obj = gameTheoryObjective(x, A)
%   x(1) = prob A uses strategy 1
%   x(2) = prob A uses strategy 2
%   x(3) = prob B uses strategy 1
%   x(4) = prob B uses strategy 2
    obj = abs(x(1)(A(1,1)x(3)+A(1,2)x(4)) + x(2)(A(2,1)x(3) + A(2,2)x(4)) - ((A(1,1)x(1)+A(2,1)x(2))x(3) + (A(1,2)x(1) + A(2,2)x(2))x(4)));
end
% Define the payoff matrix A
A = [3 0; 1 2];
% Initial guess for probabilities
x0 = [0.5, 0.5, 0.5, 0.5];
% Set bounds for probabilities (0 to 1)
lb = [0, 0, 0, 0];
ub = [1, 1, 1, 1];
% Linear equality constraints, x(1) + x(2) = 1 and x(3) + x(4) = 1
Aeq = [1, 1, 0, 0; 0, 0, 1, 1];
beq = [1, 1];
% Optimization options
options = optimoptions('fmincon','Display','iter');
% Use fmincon to solve the problem
[x,fval] = fmincon(@(x) gameTheoryObjective(x,A),x0,[],[],Aeq,beq,lb,ub,[],options);
% Display the results
disp(['Prob A strat 1: ' num2str(x(1))]);
disp(['Prob A strat 2: ' num2str(x(2))]);
disp(['Prob B strat 1: ' num2str(x(3))]);
disp(['Prob B strat 2: ' num2str(x(4))]);
disp(['Objective function value: ' num2str(fval)]);

Let’s break down this code step by step:

The gameTheoryObjective function calculates the absolute value of the difference in payoffs, as per our discussion in the previous section.
We define the payoff matrix A as per the game.
x0 is our starting guess for the optimization. It’s like picking a random starting point for fmincon.
lb and ub define the lower and upper bounds for each of our variables (probabilities).
Aeq and beq impose the constraint that the probabilities must sum to 1 for each player.
options are set to allow the display of the iterations.
Finally, fmincon tries to find the variables that minimize the objective function whilst satisfying the constraints.

4. Interpret the Results

Once fmincon finds a solution, the results indicate the optimal probabilities of each player choosing each strategy to reach the nash equilibrium. You need to interpret these values correctly in the context of your game. It’s like translating the final answer of a math problem back into the real world problem you were trying to solve.

Advanced Techniques

As you become more comfortable with fmincon, you can explore some advanced techniques:

Multiple Equilibria

Some games may have more than one Nash equilibrium. If the game has multiple equilibria, fmincon might find only one of the solutions, depending on the starting values for the probability and random elements during the optimization process. To try and find more equilibria, you should try running fmincon multiple times with different initial conditions to see if it finds different solutions.

Nonlinear Payoffs

If the relationship between a player’s choices and the payoff is not linear, you can use fmincon’s ability to handle nonlinear constraints and objective functions.

Games with More Players or Strategies

fmincon can handle games with more players, and games where each player has more than two strategies. The optimization becomes more complex, but the general method remains the same. The more strategies and players there are, the larger the vector of x, the more complex the payoff function is, and the more equality and inequality constraints are to be included.

Stochastic Games

Stochastic games include a random element. If your game has stochastic elements, you need to change the objective function to consider these random variables. You can do that by including the effects of probability distributions in your objective function.

Tips for Success

Here are a few extra tips to help you succeed when solving game theory problems with fmincon:

Start Simple: Begin with smaller and simpler games to get a feel for how the process works, and then progress to more complex games.
Visualize Your Game: Drawing out the game, or creating the payoff matrix, can help to better define the constraints and objective function.
Test Your Objective Function: Make sure your objective function is calculating what it’s meant to. Sometimes it can be difficult to spot errors. Check that it’s producing the values you expect.
Experiment with Initial Guesses: If fmincon is not finding solutions, or not finding many solutions, try different starting values. This can help fmincon find solutions in cases where it was getting stuck in local minima.
Careful with Constraints: If the constraints are not set up correctly, the solver can find incorrect solutions, or even give no solutions. Ensure all constraints are specified correctly.

By carefully formulating your optimization problem and setting up fmincon with the correct constraints and objective functions, you can solve an incredible variety of game theory problems. This is a valuable skill for anyone interested in analyzing strategic situations.

Using fmincon to solve game theory problems combines the power of mathematical optimization with the strategic insights of game theory. As you practice, you’ll become more adept at framing game theory problems as optimization problems, and fmincon can be a potent tool in your arsenal.

Matlab code for Game Theory Problem, Part- 2/2

Final Thoughts

In summary, you can solve game theory problems using fmincon in MATLAB by formulating your payoff functions and constraints. The core idea involves representing each player’s strategy as variables, and their objective function as their payoff.

Then, you would use fmincon to find a Nash equilibrium by searching for the strategies that simultaneously maximize each player’s payoff. This numerical method gives you the solution to your defined problem, demonstrating how to solve game theory problems with fmincon in MATLAB. Remember to carefully define your functions.