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