# The Winning Game

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## Objectives

### Developing the Information Technology learning strands, specifically:

• Understands and can apply payoff matrix in evaluating outcomes
• Understands and be able to explain Nash equilibium
• Understands and be able to explain dominant strategy

## Nash Equilibrium and Dominant Strategy

### Recap - Zero-sum game and Pay-off matrix

• A game is called `zero-sum` if the sum of payoffs equals zero for any outcome. That means that the winnings of the winning players are paid by the losses of the losing players. EXAMPLEs of a zero-sum games include poker, rock-paper-scissors and tennis.
• Game theory attempts to determine mathematically and logically the actions that “players” should take to secure the best outcomes for themselves in a wide array of “games.”
• A `Payoff matrix` is a table used to illustrate the player moves and the outcomes with each possible moves.

### Learn It - Payoff matrix in real life example

• Considering two toy companies, A and B. Their profits depending on their decisions on advertising or not.
• The `payoff matrix` of their decisions is as following: ### Learn It: Nash Equilibrium

• Suppose two cars are driving towards a junction from perpendicular directions
• The light is green for one and red for the other
• If the police would not ticket the drivers, would they want to break the law?
• To help you understand this scenario and possible outcomes, lets watch a short 4 minutes video:
• `A Nash Equilibrium` is a law that no one would want to break even in the absence of external force such as police in the traffic light example or in the ice cream example
• Formally, `Nash Equilibrium` is a state where no players can improve their outcomes by change of strategy as long as others remain unchanged.

• create a payoff matrix for the following scenario:

• Two cars are meeting at an intersection and want to proceed as indicated by the arrows in Figure shown below. • Each player can proceed or move. If both proceed, there is an accident. A would have a payoff of -100 in this case, and B a payoff of -1000 (since B would be made responsible for the accident, since A has the right of way). If one yields and the other proceeds, the one yielding has a payoff of -5, and the other one of +5. If both yield, it takes a little longer until they can proceed, so both have a payoff of -10.
• Analyze this simultaneous game, draw the payoff bimatrix

### Badge It Gold - Prisoners' dilemma and Nash equilibrium

• Explain Nash equilibrium using payoff matrix from Prisoners' dilemma scenario.
• Your explanation should include examining the four outcomes, A, B, C & C as shown below and why some outcomes are not Nash Equilibrium and one is. 