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Game theory is the theory of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory is mainly used in economics, political science, and psychology.

Game theory consists of two main parts: the extensive form game and the normal-form game. The normal form games are also called strategy or payoff matrices (n*n). The extensive form games include many different movements by the players through a certain state space where they interact with each other as well as with an environment through their actions that may have random results. In contrast to the normal-form games, this description uses time to make it more realistic but it cannot be solved analytically because there is no certainty over time about what will happen next or if interactions will exist.

The game-theoretic analysis can be used in a variety of fields. For example, a firm may use game theory to analyze its competition and determine what strategies will yield the greatest revenue for itself and its competitors. An evolutionary biologist might apply game theory to study an organism’s optimal aggressiveness when confronting another individual or predator. Game theoretic reasoning has proved useful in explaining human behavior as well; economic institutions, including contract law and property rights, have been described as “social games” whose payoffs depend on the actions of many individuals who are interacting strategically with each other (Sugden 1986).

In addition to having applicability in all these areas, many general results about strategic thinking have emerged from applying game theory to these contexts. For example, it has been shown that:

1. An individual’s optimal behavior depends on the other individuals’ (the “strategic interaction”), and vice versa;

2. To predict how an individual behaves in a strategic setting, you need to consider not only his utility function directly but also whether he knows what strategies are available to others and whether they know this as well; and

3. Rationality is defined in terms of payoffs, rather than in terms of what any individual actually thinks or feels; an agent is rational if his behavior brings about a good outcome given the other agents’ behavior.

Rationality in this sense is purely defined by one’s preferences and beliefs, regardless of how they are acquired. In addition to economists, game theorists include political scientists, psychologists, biologists and mathematicians as well as non-academic disciplines such as philosophy (Camerer 2003). Since there are many different kinds of models of games that vary in complexity some people who consider themselves game theorists tend to look down on others whose methods do not meet their standards. The most common approach to defining the boundary between game theory and economics was provided by Weibull: he suggested that it does not make sense to speak of game theory as a subset of economics, because game theory is not just a part of economics in the same way as general equilibrium theory or auction theory are (Weibull 2002).

The technical definition of an economic game is one where there is exactly one commodity. Thus, there can be no distinction between perfect competition and monopoly; nor can two-person relations such as marriage be considered games. Also excluded are games involving chance events whose outcomes are beyond the control of any player—the roll of dice, for example—and thus do not involve decision making under uncertainty. In addition to being basically about conflict and cooperation this approach also includes non-cooperative games like chess (if both players want the victory) but excludes cooperative games (if both players want to cooperate). The possible results of a game are usually assumed to be ranked in order of desirability: the best result is called the “utility optimum” and the worst result is called the “pareto-optimum.” In an economic game, there are at least two strategies that each player may employ; so sometimes it is helpful to think about games mathematically by treating them as vectors of length 2n.

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Game Theory is a branch of mathematics that concerns itself with the study of multiple people’s actions and behavior in strategic situations. Game theory assignments are aimed at evaluating students based on their ability to analyze a given situation, or showing how analytical skills can be applied beyond academics.

The following sections present game theory examples: some theoretical, others hypothetical; some mathematical, others purely logical; some solved as well as unsolved problems; and other typical problem-sets for students taking game theory related courses.

Within game theory assignments, one may encounter different kinds of games such as: Nash Games (aka non-cooperative games), Mixed strategies Games (where players play both cooperative and non-cooperative moves according to some pre determined rules), Cooperative Games (where players decide on the strategies together), Stackelberg Games (these are often confused with Nash games but they are not the same thing) etc….

Problem statement: Suppose we have a backgammon board whose parameter is \(N = 2 \times 2 = 4\) where a player can move his/her pieces from one point to another on the board.

Who will win? How should he or she play in order to accomplish this goal? If we assume that there is no randomness involved (i.e., weather conditions are fine, no mistakes made by both players etc.), then theoretically speaking, it is possible for any of the two players to win. This means that each of them has an equal probability of winning since they are equally capable of achieving their goal. Thus, \(p=1/2\). However in reality, there exists a certain amount of chance which indicates that some players might be more likely to win than others (e.g., if one player is more experienced or has a higher IQ). Some of the weather conditions (e.g., sun shining brightly, etc….) might also favor one player over another. This observation leads us to the following conclusion: if \(p < 1/2\), it follows that there exists some advantage for a specific player who will have an increased chance of winning over his opponent(s).

However, when both players are equally capable of winning (i.e., \(p = 1/2)\), we say that this game is fair and each move made by either player does not give him any advantage over his opponents since their chances remain equal from one turn to another. Mathematically speaking, in our example above, we modeled the situation by assuming that the die (chance) is fair and does not favor a specific move:

\(p = 1/2\) in our above example means that \(p_{1,n} = p_{2,n} \Rightarrow p_{1,i}= p_{2,i}\) for all possible moves made by players \(1\) and \(2\) respectively. Since both players are equally capable of winning, we write this inequality as follows:

where 0 < \(\beta_X \leqslant 1\) which represents the advantage of player \(X\) over his opponent. If we disregard chance effects within our game Backgammon then it is almost certain that one player will win over his or her opponent since their chances of winning (i.e., \(p = 1/2\)) are equal:

Let’s suppose there is a two-player game which will be played out between an X player and a O player on the same 3 x 3 board where each player must first select his or her move without knowing what the other one is going to do. In order to win, both players have to occupy three cells in a row where all their moves fall into four possible categories as described below: Each move made by either player can take place in any cell on the board. Within our model, we assume that each move has an equal probability 0 < p_{i,n} = p_{i,i} < 1 and where each player has an equal probability of winning

(since we do not take into account those environmental conditions which might favor one player over another) then the expected payoff for player \(X\) becomes:

Next, let us suppose that both players are equally capable of winning at a given point in time i.e., their chances of winning are both equal to 0.5 such that:

How should he or she play so as to minimize this expected payoff? Let’s start by solving the following linear optimization problem subject to the constraints described previously such that \(\gamma_X (t)\) represents the optimal choice made by player \(X\) at time \(t\):

The optimal strategy for player \(X\) is given by the linear equation above and it can be computed using any suitable numerical software package as follows:

and we get that:

\(\gamma_X (t) = \beta_{2,1} - \beta_{0,1}\). which means that if player \(X\) wants to minimize his/her expected payoff then he should always go first regardless of whether he or she wins or loses.

Then we solve the following system of equations subject to the original constraints such that A represents matrix \(A_{ij}\) with each row representing all possible moves made by player i:

\(\gamma_O (t) = A_{21} + \beta_{1,1}\) and we get that:

and we find that:

\(\gamma_O (t) = \beta_{3,2} - \beta_{0,2}\). This implies that if player \(X\) wants to minimize his/her expected payoff then he should always go second regardless of whether he or she wins or loses.

Finally, let’s look at the case where both players are equally capable of winning (i.e., their chances of winning are equal to 0.5 such that: We solve this linear optimization problem subject to the constraints described previously such that \(\gamma_X [t]\), \(\gamma_O [t]\) and \(S_{ij}\) represent the optimal choices made by player i at time \(t\), player j at t = 0 and player i respectively:

and we find that:

\(\gamma_X (t) = \beta_{3,1} - \beta_{2,0}\). which implies that if both players have a chance of winning then he or she should go first to minimize his/her expected payoff. Next, it follows from our linear optimization problem that:

where A represents matrix A with each row representing all possible moves made by player i:

This means that:

and we find that:

Finally, let’s look at the case where both players are equally capable of winning (i.e., their chances of winning are equal to 0.5 such that: We solve this linear optimization problem subject to the constraints described previously and where \(S_{ij}\) represents the optimal choice made by player i at time \(t\), player j at t = 0 and player i respectively:

and we find that:

\(\gamma_X [t] = \beta_{1,2} - \beta_{3,0}\). It means that if both players have a chance of winning then he or she should go second to minimize his/her expected payoff. Now, it follows from our linear optimization problem that:

where A represents matrix A with each row representing all possible moves made by player i:

and we find that:

which means that:

\(\gamma_O [t] = \beta_{3,2} - \beta_{1,2}\). This implies that if both players have a chance of winning then he or she should go first to minimize his/her expected payoff. Overall, we can conclude that the following theorem holds true in all cases (i.e., when p 1 > p 2 , when p 1 <>p 2 and when \(p_1\) = \(p_2)\): "If both players have an equal chance of winning then regardless of whether they win or lose player 1 should always go first to minimize his/her expected payoff."

And here is an additional example of a 2x2 game that can be solved using linear optimization:

Solve the following piecewise linear optimization problem subject to the constraints described previously such that \(A\) represents matrix \(A_{ij}\) with each row representing all possible moves made by player i at time \(t\):

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