Nash Equilibrium and Bluffing/Calling Frequencies
Introduction
In this article An introduction to game theory
 The Nash Equilibrium and its implications
 Betting and calling frequencies
Game
theory is a branch of mathematics that analyses certain kinds of conflict
situations or socalled games. In this context, the term 'game' is used for
situations in which multiple participants are competing for resources, with each
participant following a strategy, (possibly with cooperative aspects) and there
is the possibility of profit.
A central concept in game theory is the socalled Nash equilibrium, which
describes a state of strategic equilibrium between players, each player knowing
the best response to the opponent's action and no player being able to increase
their profit by making a onesided change to their strategy.
This article gives you insight into game theory, Nash equilibriums as solution
strategies, and the application of Nash equilibriums through the example of
betting and calling frequencies. Some fundamental knowledge in the area of
matrix theory is required for the comprehension of this article.
Short Introduction to Game Theory
A game in the mathematical sense is comprised of the following: a set of players
 a set containing all of a player's (pure) strategies, for each player
 a function that associates each strategy profile (that is, each player selects a strategy) with a payout tuple, which determines the payout for every player. This function determines the outcome of players selecting a certain strategy and playing.
The rest of this article will deal with the games that have two participants  a game can then very easily be represented by two m x n matrices A and B. Player 1 then has m strategies (S1,...,Sm) and player 2 has n strategies (S'1,...,S'n). If player 1 chooses the strategy Si and player 2 chooses S'j, then the payout is Aij for player 1 and Bij for player 2.
payout player 1:

S'1 
S'2 
... 
S'n 
S1 
A11 
A12 
... 
A1n 
S2 
A21 
A22 
... 
A2n 
... 
... 
... 
... 
... 
Sm 
Am1 
Am2 
... 
Amn 
The players can also use mixed strategies: this means that several pure strategies are played with certain probabilities, which, of course, add up to 100%. A mixed strategy can be represented by a vector p, with pi representing the probability of strategy number i being played.
If player 1 plays the mixed strategy p and player 2 plays the mixed strategy q, then you obtain the payouts pAq and pBq. (For pure strategies, this leads to the corresponding entry in the respective matrix.)
For player 1, strategy p is called the 'best response' to player 2 playing strategy q iff p produces maximal payout for player 1, that is, iff the following is the case:
pAq >= p'Aq for all strategies p' of player 1.
By analogy, for player 2 strategy q is called the best response to player 1 playing strategy p if the following is the case:
pBq >= pBq' for all strategies q' of player 2.
A pair of strategies (p,q), with p being player 1's strategy and q that of player 2, is called a 'Nash equilibrium (NEQ)' iff p is the best response to q and q is the best response to p.
It can be proven that there is at least one NEQ in every game (although not necessarily in pure strategies).
E An NEQ doesn't have to be Pareto optimal (an NEQ is pareto optimal if it is impossible for the NEQ to be changed so that one of the two players receives a larger payout without the other player receiving less than before.).
On the other hand, a pareto optimal pair of strategies isn't necessarily an NEQ. A wellknown example for such a pair of strategies is the prisoners' dilemma, but we will not go into details on this subject. Other equilibrium concepts that are interesting but will not be discussed here include evolutionarily stable strategies and correlated equilibrium.
If both players only have two pure strategies, then there is a very simple method for determining which strategies would produce the NEQ  the above A=B=I2 example can be used here.
Strategy p of player 1 has the form (a,1a) with a element of [0,1], while player 2's strategy has the form (b,1b) with b element of [0,1]. Now we will search for player 1's best response to strategy (b,1b) of player 2: we can do this by comparing the payouts of the strategies (1,0) and (0,1) (i.e. the payouts of pure strategies in each of the two rows).
If player 1 chooses the first row of the matrix, then his payout is b, and if he chooses the second row, then his payout is 1b. For b > 1b, which is equivalent to b > 0.5, the best response is the first row, in other words a = 1. Correspondingly, for b < 0.5 the second row is best, in other words a = 0. For b = 0,5 player 1 receives the payout of 1/2. Therefore, every strategy is a best response.
We repeat this procedure for player 2. Due to the symmetry in the example we get b=1 for a > 0.5, b = 0 for a < 0.5 and any b element of [0,1] for a = 0.5.
NEQs are defined as pairs of strategies, with each strategy being the best response to the other strategy. It is clear that in the illustration the intersections of the sets show where the NEQs are.
NEQs of this game are ((1,0),(1,0)), ((0,1),(0,1)) and ((0.5,0.5),(0.5,0.5)).