I am interested in the area of mathematics known as game theory. This sub-subject studies models used to describe human behavior, or more specifically, the strategy behind decision making. Game Theory is used in various fields such as economics, social psychology, evolutionary biology, military strategy and political science.

A common solution concept within game theory is the Nash Equilibrium. Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy.

For the formal definition of the Nash Equilibrium:

Let (S, *f*) be a game with n players, where $S_i$ is the strategy set for player i, $S=S_1 \times S_2 \times \dotsb \times S_n$ is the set of strategy profiles and $f=(f_1(x), \dotsc, f_n(x))$ is the payoff function for $x \in S$. Let $x_i$ be a strategy profile of player i and $x_{-i}$ be a strategy profile of all players except for player i. When each player i $\in \{1, \dotsc, n\}$ chooses strategy $x_i$ resulting in strategy profile $x = (x_1, \dotsc, x_n)$ then player i obtains payoff $f_i(x)$. Note that the payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile $x^* \in S$ is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is

$\forall i,x_i\in S_i : f_i(x^*_{i}, x^*_{-i}) \geq f_i(x_{i},x^*_{-i})$.

When the inequality above holds strictly (with > instead of ≥) for all players and all feasible alternative strategies, then the equilibrium is classified as a strict Nash equilibrium. If instead, for some player, there is exact equality between $x^*_i$ and some other strategy in the set S, then the equilibrium is classified as a weak Nash equilibrium.

A game can have either a pure-strategy or a mixed Nash Equilibrium.

The article What is Game Theory? from the Economic Times is a good brief, general introduction into game theory.

Samuel CoskeyAwesome post! One thing I’ve always wondered is: In practice, should people always play by the Nash Equilibrium strategy (when there is one)? Or can there be other sorts of equilibrium-y strategies that can be advantageous as well?

Marc GarlandUsing mathematics to describe human behavior sounds very useful. That sounds particularly complex due to how different everyone is from one-another. An interesting and challenging topic, I’m sure.

I have never seen the Nash Equilibrium. That is incredible! And complicated…. I want to know more. Cool post!

tylermurphyThis is a such a fantastic field of study. Have you talked to Dr. Scheepers here on campus? He specializes in game theory. We are trying to get enough students interested to sign up for a special topics course in game theory this spring. I admit to not knowing much about game theory, though I am interested in infinite games. Does the Nash Equilibrium hold up in an infinite game? Or a game with infinite players?

carriesmithI have always been interested in human behavior and to now discover that there is a theory behind the way we make decisions has left me intrigued. I have been thinking about the difference between logic based thinking and emotional based thinking and why a person would choose one over the other. If you think about it, decision making plays a huge part in the evolution and it would be incredibly interesting to have a better understanding when it comes to the “how” and “why” we humans make decisions. This subject seems very interesting and I would love to explore it further! Thank you so much for sharing Janae!