The Situation: ◦ Ben and Jerry are opening a new ice cream shop. ◦ You agree on everything except your elevation preferences are diametrically opposed: BenJerry Prefers low elevation “The lower the better!” Prefers high elevations “The higher the better!”
Your Task: ◦ To maximize the number of customers you agree that the ice cream shop should be at the intersection of a route (A,B,C) and a highway (1,2,3). ◦ To determine the final location Ben will select the highway and Jerry will simultaneously choose a route. Route B Route Route C
[Blank slide to thwart the smartboard from giving away the answer]
Three-dimensional road map of possible choices Ben Highways (Wants low) Routes123 Row Minima Jerry Routes (Wants high) A10464 B6595 C2372 Column Maxima 1059
In the above game the numbers in the middle represent the “batting averages” for the batter against the pitcher based on the pitch selected and the swing selected (.3 is a hit 30% of the time) A) What is the maximin of this scenario? B) What is the minimax of this scenario? C) Does a saddle point exist? If not, what is the gap between the minimax and the maximin? Baseball duel (2-player game) Pitcher FastballCurve Batter Fastball0.3000.200 Curve0.1000.500
What is the expected batting average? ◦ Expected Value: When a player resorts to a mixed strategy, the resulting outcome of the game is no longer predictable. Instead, the outcome must be described in terms of weighted probabilities. We are essentially splitting up the gap between the maximin and minimax between the 2 players
Cory MatthewsShawn Hunter JaimieMasha JamieAmy TayoLisa AbbasSarah MattKaty JTChad ( Whoever’s partner bailed on them) Last week Dr. Feeney’s glasses were stolen after-class.
A two-person variable sum game Each player has two strategies Deny (cooperate with other player) Confess on partner (defect against other player) Mutual defect is always worse than mutual cooperation (i.e. both confessing on the other is worse than both denying) (Snitch)
When each person selects their own best individual strategy, both people suffer in the end. ◦ For both Shawn and Cory snitching strategy dominates denying ◦ But if both snitch, it’s worse than if both deny For the best mutual outcome to be reached, cooperation is needed.
Scenario 1: Individual Strategies Scenario 2: Cooperation Corey ConfessesShawn Confesses Shawn Denies Both Receive 2 Weeks Both Receive 1 Week In scenario where each selects their own best individual strategy, both suffer in the end.
There are four possible outcomes: Player 1 accelerates, Player 2 swerves Player 1 wins, both live Player 2 accelerates, Player 1 swerves Player 2 wins, both live Player 1 swerves, Player 2 swerves Both players lose-face, but both players live. Player 1 accelerates, Player 2 accelerates Neither technically wins, both players die. Catastrophic outcome
Activity: Assign point values to each of these outcomes in the table on your handout using values from 1-10 (with 1 being the worst).
Situation Each person has one bullet. Each decides whether to shoot or not shoot simultaneously Two goals: #1 Survive #2 Kill as many others as possible *What should be the expected outcome?*
How would the scenario be different if all decisions were not made simultaneously and all players must fire their gun? Key Points to Remember: ◦ There is only 1 bullet in each gun ◦ A decides, then B, then C (but even if A shoots B, B still gets a shot in this scenario- simultaneous sequential) ◦ What should be the expected outcome?*
How would the scenario be different if all decisions are not made simultaneously and players are not required to fire their gun? ◦ Note: If you fire your gun you must shoot someone *What should be the expected outcome?* Real Life Application: Is a truel mathematically a better model than a duel when aiming to prevent conflicts?
Which of these activities could you use in your classroom? ◦ What would you modify?