3 – The Winning Game

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

• 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.

• 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:

• 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.
• Analyse this simultaneous game and draw the payoff matrix.

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

Conduct some research online and write 200 words on:

• What a dominant strategy is?
• Find an example of dominant strategy explain how the dominant strategy works in your example.