# Structure of a game

Game theory is based on the concept of strategy and payoffs.

- Strategy indicates an action that a player takes when challenged to solve a particular problem.
- Payoff refers to the outcome of the strategy applied by a player. The payoff may depend on more than one player like in the infamous prisoner's dilemma.

## Cooperative and non-cooperative

In *Cooperative games*, players are convinced to adopt a particular strategy through negotiations and agreements between players. In the prisoner's dilemma, had the players been able to contact each other, then they must have decided to remain silent. Therefore, their negotiation would have helped in resolving the problem.

## Normal and extensive form

*Normal form games* refer to the description of game in the form of matrix. In other words, when the payoff and strategies of a game are represented in tabular form (demonstrating the strategies adopted by the different players and their possible outcomes), it is called a *normal form game*. *Normal form games* help in identifying the dominated strategies and Nash equilibrium.

*Extensive form games* refer to the description of a game in the form of a decision tree (a tree-like structure in which the names of players are represented on different nodes and feasible actions and pay offs of each player are also “a given”). Extensive form games help in the representation of events that can occur by chance.

## Simultaneous and sequential moves

In *Simultaneous games*, the move of two players (and strategy adopted) is simultaneous: Players do not have knowledge about the move of other players.

Suppose there are only two oil producers in the world. Both can make profits of 30 million. If one produces a lot and the other average, the one that produces a lot will earn 25 million.

Production | Average | A lot |
---|---|---|

Average | 50,50 | 25,55 |

A lot | 55,25 | 30,30 |

This is a *simultaneous game*, as both actors make their decisions at the same time. Each actor's dominant strategy is to produce a high level of oil instead of an average production, and they will each make $30 million per month, and start dreaming about cartels.

In *sequential games* players are aware about the moves of players who have already adopted a strategy, but do not have a deep knowledge about the strategies of other players. For example, a player has knowledge that the other player would not use a single strategy, but is not sure about the number of strategies the other player may use.

*Simultaneous games* are represented in *normal form* and *sequential games* are typically represented in *extensive form*.

## Constant, zero, and non-zero sum

In *constant sum games*, the sum of outcome of all the players remains constant even if the outcomes are different.

*Zero sum games* are *constant sum games* in which the sum of outcomes of all players is zero: The strategies of different players cannot affect the available resources, and the gain of one player is always equal to the loss of the other player. For example, Rock-Paper-Scissors (RPS) is a *simultaneous zero sum game*. The only equilibrium is in mixed strategies. Each player plays each strategy with equal probability, resulting in an expected payoff of zero for each player.

To determine who is required to do some “thing”, two decision makers simultaneously make one of three symbols with their fists - a rock, paper, or scissors. Simple rules of “rock breaks scissors, scissors cut paper, and paper covers rock” dictate which symbol beats the other. If both symbols are the same, the game is a tie.

Rock | Paper | Scissors | |
---|---|---|---|

Rock | 0,0 | -1,1 | 1,-1 |

Paper | 1,-1 | 0,0 | -1,1 |

Scissors | -1,1 | 1,-1 | 0,0 |

*Non-zero sum games* are games in which sum of the outcomes of all the players is not zero. *Cooperative games* are examples of *non-zero games*.

## Symmetric and asymmetric

In *symmetric games*, strategies adopted by all players are the same. The decisions depend on the strategies used, not on the players of the game. Even when interchanging players, the decisions remain the same. An example is the prisoner's dilemma.

In *asymmetric games*, strategies adopted by players are different. A strategy providing benefit for one player is not necessarily equally beneficial for the other player. Decision making in asymmetric games depends on the different types of strategies and decision of players.

## Mixed strategies

A *mixed strategy* is a collection of moves together with a corresponding set of weights which are followed probabilistically in the playing of a game.

Every finite, zero-sum, two-person game has optimal mixed strategies.

## Minimax

*Minimax* is a simple rule that can be applied to everyday situations: *pick the strategy where the maximum advantage of your opponent is minimised*. Minimax assumes perfect information. In real life, things are **somewhat** more confusing, not least because our negotiating adversary is rarely limited to two strategies.

## N-person games

Games with more than two people are called N-person games:

- All N-person games are defined as zero sum.
- Most solutions are based on voting models and coalition building.
- Your power is derived from the probability of casting the 51st vote out of 100.
- Size isn’t always an advantage. If two big players are antagonists, the smallest player can call the shots!