Post on 16-Dec-2015
Game TheoryThursday, May 6th
Location GameBen Jerry
Jaimie Amy
Jamie Masha
Tayo Lisa
Abbas Sarah
Chad Katy
JT Matt
The Situation:◦ Ben and Jerry are opening a new ice cream shop.
◦ You agree on everything except your elevation preferences are diametrically opposed:
Location Game
Ben Jerry
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.
Location Game
Route B
Route
Route C
[Blank slide to thwart the smartboard from giving away the answer]
The SolutionThree-dimensional road map
of possible choicesBen Highways (Wants low)
Routes 1 2 3Row
Minima
Jerry Route
s (Wants high)
A 10 4 6 4
B 6 5 9 5
C 2 3 7 2
Column Maxima
10 5 9
Players? –
Options? –
Strategies? -
Outcome? -
Location game wrap-up using Game Theory terminology
Jerry has all of the candy, this time Jerry picks a column, and Ben simultaneously picks a row. The intersecting number is the number of candies that Jerry gives Ben.
You Try!
3 7 2
8 5 1
6 9 4
Game TheoryFriday, May 7th
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?
Do NowBaseball duel (2-player game)
Pitcher
Fastball Curve
BatterFastball 0.300 0.200
Curve 0.100 0.500
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
Baseball Duel Baseball duel (2-player game) with probabilities
PitcherF C
F 0.300 0.200 qC 0.100 0.500 1-q
p 1-p
Batter
Activity
Cory Matthews Shawn Hunter
Jaimie Masha
Jamie Amy
Tayo Lisa
Abbas Sarah
Matt Katy
JT Chad ( Whoever’s partner bailed on them)
• Last week Dr. Feeney’s glasses were stolen after-class.
SentencesCory Matthews
Decision Sentence
Shawn Hunter
Decision Sentence
Jaimie Masha
Jamie Amy
Tayo Lisa
Abbas Sarah
Matt Katy
JT Chad ( )
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)
Prisoner’s Dilemma
(Snitch)
(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.
Nash Equilibrium exists when…
Nash Equilibrium Example…The Communication Factor
Scenario 1: Individual Strategies
Scenario 2: Cooperation
Corey Confesses Shawn Confesses Shawn Denies 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
The Game of Chicken
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) .
The Game of Chicken
Situation Each person has one bullet. Each decides whether toshoot or not shootsimultaneously Two goals: #1 Survive
#2 Kill as many others as possible
*What should be the expected outcome?*
A Truel
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?*
Truel Extension – Scenario #2
Truel Extension – Scenario #3 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?
Closing-Classroom applications