The Volunteer's Dilemma describes a situation where each player can either volunteer or defect. If at least one player volunteers, all other players marginally benefit from defecting. If no player volunteers, all players have a very low payoff. Generate a Volunteer's Dilemma game with a volunteer and a detractor:

Find the expected payoffs with a likely volunteer and an unlikely volunteer:

Find the expected payoffs with likely volunteers:

Find the expected payoffs with unlikely volunteers:

The Discoordination game is a hybrid form of coordination and anti-coordination games, where one player's incentive is to coordinate, while the other player tries to avoid this. Generate a Volunteer's Dilemma game with a volunteer and a detractor:

Find the expected payoffs with cooperative players:

Find the expected payoffs with uncooperative players:

Find the expected payoffs with a cooperative player and a uncooperative player:

The Cournot Oligopoly game describes a situation where a group of firms produces the same good. Each firm must consider the production cost and the quantity that the other firms are producing. Only the firms with the lowest price sell goods. Generate a Cournot Oligopoly game:

Find the expected payoffs when the first two players collude:

A price war refers 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 3 firms where each firm has a choice between a low price and a high price:

Visualize the game:

Clearly, cooperating players have an interest in choosing high prices. Show this by comparing the payoffs of multiple strategies:

The Colonel Blotto game describes a situation where officers (players) are tasked to simultaneously distribute limited resources over several objects (battlefields). The player devoting the most resources to a battlefield wins that battlefield, and the gain (or payoff) is equal to the total number of battlefields won. Generate a Colonel Blotto game:

Find the expected payoffs for a 50-50 strategy:

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, taking 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 time taken for each case:

Maximize the negative commute time:

Create a traffic game based on the payoff matrix:

Visualize the traffic game:

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

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

Following the traffic game above, you can automate the modeling and solving for a general road network. Here start and end are the 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):

Consider the expected payoff for each player in a two-person game:

Find the expression for each player payoff:

Assume that each player changes their expected payoff following the gradient:

Plot the gradient as a stream plot:

From this plot, you can see that only the pure strategies {{0,1},{1,0}} and {{1,0},{0,1}} are "stable" in the sense that the trajectories converge to them: