Create a zero-sum two-player game:

For zero-sum two-player games, for any payoff p_i,j for player 1, the payoff for player 2 is -p_i,j:

Create a two-player game:

Show the dataset:

Create a two-player game with action labels and player labels:

Show the dataset:

Represent a Rock Paper Scissors game as a directed graph:

Create a Rock Paper Scissors game:

Plot it:

Create a zero-sum two-player game with a directed complete graph:

Show the dataset:

Create a zero-sum two-player game using array rules:

Show the plot:

Create a symmetric two-player game based on the matrix

using array rules:

Show the plot:

Manually create a composite array based on a component array:

Create a composite array based on a component array using MatrixGame:

Manually create a component array based on a composite array:

Create a component array based on a composite array using MatrixGame:

MatrixGame objects include properties that give information about the game:

The "Properties" property gives a list of properties available:

The "ObjectType" property gives the object type of the game:

The "Summary" property gives a brief summary of information about the game:

The "CompositePayoffArray" property gives the composite form of the payoff array of the game:

The "ComponentPayoffArray" property gives the component form of the payoff array of the game:

The "PayoffArray" property also gives the component form of the payoff array of the game:

The "PayoffData" property gives the list of possible payoffs for each player:

The "Dataset" property gives the matrix game as a dataset of payoffs:

The "Plot" property plots the matrix game in the same fashion as MatrixGamePlot:

The "TreeGame" property converts any MatrixGame into a TreeGame:

Specify the names of the actions of each player:

These can be accessed through GameActionLabels:

This changes the result from the "Dataset" property:

Consider a random game about the yield of vegetables starting with "p":

View the dataset:

Note if the wrong number of action labels is provided, they will be ignored:

Specify the names of the actions of each player:

These can be accessed through GamePlayerLabels:

Consider a random game about the yield of vegetables for three farmers:

View the plot and note that the tooltips of the figure were based on the player labels:

The common side-blotched lizards have three distinct male mating strategies. The orange-necked ones maintain large harems, the blue-necked ones keep fewer females but are more protective, and the yellow-necked ones pretend to be females and can sneak into harems. Based on field observations, the fitness of each individual male strategy is known, which strongly depends on the already-present males' frequencies. gives the fitness of the strategy in an environment dominated by males with the strategy:

Assuming the game is symmetric, find the component payoff array:

Define a symmetric two-player game:

This game is similar to Rock Paper Scissors, as can be seen from their plots:

In a mixed population, the fitness of each strategy can be determined as the expected payoff of the mating game:

Assuming that the population of each strategy in the next generation is proportional to the current population and its fitness yields the following evolutionary dynamics:

Starting from 35% yellow-, 50% blue- and 15% orange-necked lizards, an oscillatory pattern can be observed in the first 100 generations:

Find the Nash equilibrium:

The oscillation may be avoided when starting at the Nash equilibrium:

Rock Paper Scissors is based on a cyclic order rock>scissors, scissors>paper, paper>rock. Create the graph representation of this game:

Create the game using the graph representation:

Show the dataset:

Compute the Nash equilibrium:

Generalize Rock Paper Scissors to Rock Paper Scissors Fire Water by setting up a dominance order using a graph:

Create the whole game as a zero-sum game:

Show the dataset:

Compute the Nash equilibrium:

Morra is a hand game that dates back thousands of years to ancient Roman and Greek times. Each player simultaneously reveals their hand, extending any number of fingers, and calls out a number. Any player who successfully guesses the total number of fingers revealed by all players combined scores a point.

A two-player, "m-finger" Morra game can be constructed in the following way:

A traditional five-finger Morra is a symmetric zero-sum game with 50 available actions for both players:

In various games, including soccer, baseball and tennis, players face decisions where no predetermined action is optimal. For example, in soccer during penalty kicks, both the kicker and the goalkeeper should randomize if they kick/move to the right or the left. Because of missed kicks and other errors, the scoring probabilities can vary. Based on real-world data, a realistic scoring matrix looks like the following:

An obvious question is how the kicker and the goalkeeper should randomize their strategies:

And the scoring frequencies in this case:

One can carry out the same computations when the scoring probabilities are changed:

Interestingly, the conventional wisdom to "prioritize your strong side" is not optimal in most of the cases:

Three hungry men go to a restaurant and decide to split the bill evenly. There are two meal options, Cheap and Expensive. The Cheap meal is fairly priced at $10, while the Expensive meal would be fairly priced at $40 but is overpriced at $50. Represent this situation as a MatrixGame:

Plot the game:

Find the Nash equilibrium:

A price war refers to a game where multiple firms have an interest in offering the lowest price, but the payoff of any firm is directly correlated to the price chosen. Consider a price war between three firms where each firm has the choice between a low price and a high price:

Visualize the game:

Despite the common interest of having the price as high as possible, competition creates a Nash equilibrium at the low price:

A couple can choose from activities A, B, C. The wife prefers A, the husband B, and neither of them likes C. Represent the situation using a MatrixGame:

Plot the game:

Find the Nash equilibria:

The Colonel Blotto game is a two-person zero-sum game in which the players are tasked to simultaneously distribute limited troops over several battlefields. The player devoting the most troops to a battlefield wins that battlefield, and the payoff is equal to the total number of battlefields won. Consider a Colonel Blotto game with two battlefields where Colonel Blotto has 2 units and Enemy has 4 units:

Plot the game:

Find some of the Nash equilibria:

Find the payoffs of each of those strategies:

Two players drive up to the same intersection at the same time. Each player can either cross or stop. If both attempt to cross, the result is a fatal accident. Successfully crossing has a payoff of 1, and stopping has a payoff of 0. Represent this situation as a MatrixGame:

Generally, strategies excluding the fatal accident are Nash equilibria:

Amazingly, adding one or more roads to a road network can slow overall traffic, known as Braess's paradox. Given a road network from a to b, take two different paths either via c or d. The roads ac and db take 20 n minutes, where n is the number of drivers on the road; cb and ad take 45 minutes independent of the number of drivers. Represent the road network as a graph:

For two drivers n=2, find all paths from a to b:

Count the number of drivers for each choice of path, which gives four cases:

Find the total the time taken for each case:

Maximize the negative commute time:

Create the traffic game based on the payoff matrix:

Visualize the traffic game:

The two pure strategies correspond to the commuters picking different paths:

Those pure strategies also give the shortest commute time of 65 minutes:

Following the traffic game above, the modeling and solving for a general road network can be automated. Here start and end are start and end vertices, and each road segment capacity is encoded using edge weight:

Add a short high-capacity road segment between d and c:

This situation defines a new traffic game for two drivers:

Visualize the traffic game:

Solve the game:

The new possibility of switching between the two previously existing routes resulted in a Prisoner's Dilemma–like situation, where the only stable solution results in a longer commuting time (82 minutes), compared to the restricted network (65 minutes):