**Mathematical Models**

When we repeat these kind of experiments we start to create mathematical models of patterns of behavior. Mathematical models are very important in game theory and in video game design. We use them to create complex social and combat mechanics and AI interactions. We use them to create strategy scenarios in Real-Time Strategy games. And we use them to balance PvP strategy games of all sorts.

It really is a math equation with a human component. Mathematics is core to almost all game design as it relates to game theory. For the purposes of this course it is not necessary to display and explain all of the different ways to model game theory scenarios. It is, after all, not a math course. Rather we will seek to understand the overall concepts and allow those who want to, to further explore the detailed math models. What we will focus on is the concepts of strategy in game theory and how it relates to game design.

**The Science of Strategy**

Can people’s thought processes and actions be reduced to a set of probability numbers like crafting the wing of an airplane? Do people function like foreseeable algorithms to the point of accurate predictability? In this case, we can only focus on two people where the “winner” only gains what the “loser” loses. It does not consider a host of other possible variations in human behavior, complexity, experience, or motivation. **The truth is most things are much more complex than the Prisoner’s Dilemma. **What game theory really does is look at how people choose solution strategies that have defined value over other solutions.

**The Nash Equilibrium **

In 1994, mathematician John Nash, who was the subject of the movie, A Beautiful Mind, won a Nobel Prize for his work on economics using game theory. Nash developed solution concepts as formal rules of how strategies will be adopted by people in states of competition. Specifically, the Nash Equilibrium states that each player is assumed to understand the optimal strategies of all other players, and that no one will gain by one competitor changing only their strategy.** In other words, if all players have adopted a strategy that is the best strategy in relation to others strategies, we have an equilibrium.** Like the perfectly balanced game paper, rock, scissors – each strategy dominates another one and is countered by another as well. In a video game, this means a balance of strategies among all players, which is essential for good game design.

#### Optimum and Dominant Strategies

**Optimum Strategies **

The Prisoner’s Dilemma does provide a simple basis to understand the concept of game theory as we add more complexity. It gives us baseline to identify optimum strategies. With the information to determine the best course of action optimum strategies are formed in the face of like competition. **In kindergarten, we learned the Golden Rule: **Treat others the way you would like to be treated. That is a statement that fits game theory as well: making choices that consider the strategy of others is a solid strategy.

**Dominant Strategies**

Looking again at the Prisoner’s Dilemma, we see that the optimum strategy for the accused would be for both to refuse to testify ( to treat the other prisoner how they would want to be treated – with cooperation and zero punishment ). Game Theory, however, suggests each person will instead choose a dominant strategy that is better for them. That is, the decision that has a strictly higher payoff for each prisoner’s own self-interest separate from the other prisoner’s self-interest. This, of course, is actually a poor strategy as it insures a less than optimum outcome for both prisoners.

The problem with dominant strategies in game design is that they present such a compelling choice; there is no reason to try any other strategy. They eliminate all other options as useless. **If a game presents two strategies for victory, and one always dominates over the other, we might as well just remove the lesser strategy from the game. What we really want is mixed strategies, where no one strategy dominates all of the other options. This is not immediately intuitive and so it will take practice to get good at eliminating dominant strategies.**