Module 12.3 Nash Equlibrium Pt. 2 Lecture

Overview of the Game Theory Concept

• The discussion centers on Nash equilibria, which are points in games where players do not have incentives to change their strategies given the strategy of the other player.

First Game: Alyssa and Becky

• Players: Alyssa (does not want to be with Becky) and Becky (wants to be with Alyssa).

• Locations: MSC and Northgate.

• Payoff Matrix:

  • Both at MSC:

  • Becky's payoff: 10

  • Alyssa's payoff: -3

  • Both at Northgate:

  • Becky's payoff: 10

  • Alyssa's payoff: -3

  • Alyssa at MSC, Becky at Northgate:

  • Becky's payoff: 0

  • Alyssa's payoff: 12

  • Alyssa at Northgate, Becky at MSC:

  • Becky's payoff: 12

  • Alyssa's payoff: 0

• Finding the Nash Equilibrium:

  • Starting both at MSC: Alyssa prefers to switch to Northgate (payoff 12 vs -3).

  • Both in Northgate: Alyssa prefers to be at MSC because her payoff is higher.

  • This back and forth means they are not reaching a Nash equilibrium because neither stabilizes their choices.

• Conclusion:

  • There's no Nash equilibrium in this scenario. Both players continuously prefer to switch their choices.

Second Game: Yolanda and Zach

• Players: Yolanda (likes MSC) and Zach (likes Northgate).

• Payoff Matrix:

  • Both at MSC:

  • Zach's payoff: 3

  • Yolanda's payoff: 6

  • Yolanda at Northgate, Zach at MSC:

  • Zach's payoff: -4

  • Yolanda's payoff: 5

  • Both at Northgate:

  • Zach's payoff: 12

  • Yolanda's payoff: 15

  • Yolanda at MSC, Zach at Northgate:

  • Zach's payoff: 8

  • Yolanda's payoff: -1

• Finding the Nash Equilibrium:

  • Start at MSC: Zach wants to switch to Northgate (higher payoff).

  • If both switch to Northgate, Zach has a payoff of 12 and Yolanda has a payoff of 15.

  • This means if they are both at Northgate, neither wants to change their strategy, achieving stability.

• Conclusion:

  • The Nash equilibrium is when both players are at Northgate (payoff 12 for Zach and 15 for Yolanda).

Extensions

• Nash equilibria can be extended to games with more players and strategies.

• The payoff matrix increases in size (e.g., 3x3) but the approach to find equilibria remains the same:

  • Check square for stability: If no players want to change strategies, it is a Nash equilibrium.