# Game Theory

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Updated on Wednesday, May 03, 2017

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Game theory analyzes strategic interactions in which the outcome of one's choices depends upon the choices of others. For a situation to be considered a game, there must be at least two rational players who take into account one another's actions when formulating their own strategies.

If one does not consider the actions of other players, then the problem becomes one of standard decision analysis, and one is likely to arrive at a strategy that is not optimal. For example, a company that reduces prices to increase sales and therefore increase profit may lose money if other players respond with price cuts. As another example, consider a risk averse company that makes its decisions by maximizing its minimum payoff (maxmin strategy) without considering the reactions of its opponents. In such a case, the minimum payoff might be one that would not have occurred anyway because the opponent might never find it optimal to implement a strategy that would make it come about. In many situations, it is crucial to consider the moves of one's opponent(s).

Game theory assumes that one has opponents who are adjusting their strategies according to what they believe everybody else is doing. The exact level of sophistication of the opponents should be part of one's strategy. If the opponent makes his/her decisions randomly, then one's strategy might be very different than it would be if the opponent is considering other's moves. To analyze such a game, one puts oneself in the other player's shoes, recognizing that the opponent, being clever, is doing the same. When this consideration of the other player's moves continues indefinitely, the result is an infinite regress. Game theory provides the tools to analyze such problems.

Game theory can be used to analyze a wide range of strategic interaction environments including oligopolies, sports, and politics. Many product failures can be attributed to the failure to consider adequately the responses of competitors. Game theory forces one to consider the range of a rival's responses.

## Elements of a Game

- Players: The decision makers in the game.
- Actions: Choices available to a player.
- Information: Knowledge that a player has when making a decision.
- Strategies: Rules that tell a player which action to take at each point of the game.
- Outcomes: The results that unfold, such as a price war, world peace, etc.
- Payoffs: The utilities (or happiness) that each player realizes for a particular outcome.
- Equilibria: An equilibrium is a stable result. Equilibria are not necessarily good outcomes, a fact that is illustrated by the prisoner's dilemma.

## Game Theory Framework

When evaluating a situation in which game theory is applicable, the following framework is useful.- Define the problem
- Identify the critical factors. Examples of critical factors include differentiated products, first-mover advantage, entry and exit costs, variable costs, etc.
- Build a model, such as a bimatrix game or an extensive form game.
- Develop intuition by using the model
- Formulate a strategy - cover all possible scenarios.

A good strategy could be used as a set of instructions for someone who knows nothing about the problem. It specifies the best action for each possible observation. The best strategy may be formulated by first evaluating the complete set of strategies. The complete set of strategies is a list of all possible actions for each possible observation.

## Bimatrix Games

In a bimatrix game, there are two players who effectively make their moves simultaneously without knowing the other player's action. A bimatrix game can be represented by a matrix of rows and columns. Each cell in the matrix has a pair of numbers representing the payoff for the row player (the first number) and the column player (the second number). The game has the following form:Row Player (RP) |
Column Player (CP) | |

CP Option 1 | CP Option 2 | |

RP Option 1 |
(Payout to RP, Payout to CP) | (Payout to RP, Payout to CP) |

RP Option 2 |
(Payout to RP, Payout to CP) | (Payout to RP, Payout to CP) |

The general form of equilibrium in a bimatrix game is called a Nash Equilibrium. If both rivals have dominant strategies that coincide, then the equilibrium is called a dominant strategy equilibrium, a special case of a Nash equilibrium.

A dominant strategy, if it exists, is for one of the players the strategy that is always the best strategy regardless of what one's rival plays. A dominated strategy is one that is always the worst regardless of what one's rival plays. In games having more than two rows or problems, one may find it useful to identify one option that is always better or worse than another option, in other words, that dominates or is dominated by another option. In this case, the inferior strategy can be eliminated and the game simplified such that more options can be eliminated based on the smaller matrix.

If no options dominate any others, a Nash equilibrium might still be found by evaluating each player's best option for each option of the opponent. If a cell coincides for both players, then that cell is a Nash equilibrium. A game can have more than one Nash equilibrium, but one of them may be the more likely outcome if it is better for both players.## Extensive Form Games

Extensive form games are modeled with dots with arrows that point to other dots. A node is a decision point. The beginning point is depicted by an open dot, which usually represents a state from which a situation will arise by chance. Decision points are labeled with the name of the player making the decision. The following diagram shows the structure of an extensive-form game representation.Structure of an Extensive Form Game

Chance nodes can appear anywhere in the extensive form tree.

An information set is a collection of nodes that are controlled by the same player, but which are indistinguishable for that player. In other words, for nodes in the same information set, the player does not know which one he/she is at, but does know that these nodes are different. In the preceding diagram, the drawn information sets might arise if Decision A and Decision C were indistinguishable to Player 2, as well as Decision B and Decision D. If a single dotted line encompassed all the Player 2 decision nodes (or 4 dotted circles all connected), then Player 2 would not be able to distinguish between any of the four decisions.

An extensive form game without information sets designated is one in which the players know exactly where they are in the tree. This situation is equivalent to one of dotted circles drawn around each decision point in the tree but not connected to one another. If neither player can observe anything about the other player's action, the sequential extensive form game can be reduced to the simultaneous-action bimatrix game.

## Strategic Form Game Representation

The extensive form of representing a game can become difficult to manage as the game gets larger, and the Nash equilibria may become difficult to find. The extensive form representation can be collapsed into the normal form, which encodes the game into a strategy that describes the action to take for each conceivable situation (for example, for each information set). The normal form is a complete listing of all the possible strategies along with their payoffs. For a generic case in which there are three situations (information sets) based on Player 1's move, and two possible responses by Player 2, the normal form takes the following structure:Player 1 | |||||||

If Player 1 does A, then |
If Player 1 does B, then |
If Player 1 does C, then |
A | B | C | ||

Player 2 | 1) | d | d | d | (__,__) | (__,__) | (__,__) |

2) | d | d | e | (__,__) | (__,__) | (__,__) | |

3) | d | e | d | (__,__) | (__,__) | (__,__) | |

4) | d | e | e | (__,__) | (__,__) | (__,__) | |

5) | e | d | d | (__,__) | (__,__) | (__,__) | |

6) | e | d | e | (__,__) | (__,__) | (__,__) | |

7) | e | e | d | (__,__) | (__,__) | (__,__) | |

8) | e | e | e | (__,__) | (__,__) | (__,__) | |

Payoffs |

Signaling and Threats

A signal is a commitment that changes the strategic situation. For example, if there exists danger of a price war, a smaller firm may sell a plant in order to reduce capacity and limit the amount of a larger firm's demand that it can steal, signaling that it does not intend to build substantial market share. As another example, a sports player who for some reason refuses to undergo an medical exam before negotiating his salary will raise doubts as to whether he has a medical condition that might affect his performance. This player might wish to signal that he does not have such a condition by proposing that his pay be tied to his performance. For a signal to be effective, its cost to a bluffer must exceed the benefits.

A threat is credible if it is believable. A threat is believable if it is in the best interest of the one making the threat to carry it out.

Auctions

Game theory can be used to analyze strategies for auctions. Auctions are useful mechanisms for price determination. Common auction formats include the English auction, Dutch auction, and sealed bid auction.

Legal Issues

One must consider that just because a possible action of one's opponent is illegal, this technicality might not prevent the action from being taken. For example, if a small but growing company's opponent is contemplating an illegal action, one should not rule out the possibility of that action without considering the cost of filing an anti-trust suit. The time and money required might not be the best thing for the development of the business. Even if the opponent is found guilty, the small company may be out of business by the time the suit is resolved. On the other hand, the publicity from such a lawsuit might be exactly what the smaller player needs to build brand awareness and win over new customers. (This is an example of exploitation of a big versus small asymmetry.)

#### Summary

Game theory helps one to develop optimal strategies. In an environment in which many outcomes are pre-determined when sophisticated players follow their best strategies, the way to improve one's payoff is to change the actual structure of the strategic interactions before the game is even played.

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