Extensive Form Games 2/10/17 Alpha-Beta Pruning Exercise + - - PowerPoint PPT Presentation

Extensive Form Games

2/10/17

Alpha-Beta Pruning Exercise

+ − − + + + + + + − − − − − − − − − − 5 6 7 4 5 3 6 8 6 9 4 7 6 9 9 8 6 2 5

What can we model so far?

With minimax, we can solve:

• two-player, zero-sum, complete information,

sequential move games

• lots of classic board games: chess, checkers, connect 4…
• not much else, and the search space for the above is
• ften too big

With backwards induction (so far) we can solve:

• complete information, sequential move games
• simple models of corporate competition, a few other

economic applications

Example Application: Resource Sharing

Also known as cake-cutting, as in “I cut, you choose”.

• One agent proposes a division of a desirable resource, the
• ther accepts or rejects that division.
• Here, we model the agents as getting utility +1 for each unit
• f resource, but they’re also spiteful so they feel a disutility
• f 0.5 if they receive less than half.
• MANY variations on this model are possible.

1 2 2 2 2 3,-.5 0,0 2,.5 0,0 .5,2 0,0

• .5,3

0,0 3,0 2,1 1,2 0,3 A R A R A R A R

What can’t we model so far?

Random Outcomes Simultaneous Moves Incomplete Information

How can we handle these cases?

Random Outcomes

• Moves by “nature”
• Compute expected value

Incomplete Information

• Information sets
• You could be at several nodes, but don’t know which

Simultaneous Moves

• Normal form games
• Mixed strategies (behaving randomly on purpose)

Randomness: Moves by nature

1 2 1 1 2 2 N

• 76,-12
• 9,57
• 31,53

12,-42 29,-30

• 11,73

31,3 8,24

• 9,57

12,-42

• 9,57
• 11,73

8,24 .4 .6

Compute expected values:

• 11*.4 + 8*.6 = 0.4

73*.4 + 24*.6 = 43.6

0.4,43.6 0.4,43.6

Incomplete Information

2 1 1 N 2 2 1

Information Set: A set of decision nodes among which a player can’t distinguish.

56,91 68,54 67,49 58,73 39,25 23,33 69,1 20,7 .8 .2

Simultaneous Moves

1 2 2 2

0,0

• 1,1

1,-1

• 1,1

0,0 1,-1 1,-1

• 1,1

0,0 P R S P R S P R S P R S

R P S R 0,0

• 1,1

1,-1 P 1,-1 0,0

• 1,1

S

• 1,1

1,-1 0,0 2 1

Optimal play may require randomizing your action to avoid predictability. Key idea: mixed strategy Nash equilibrium

Example: Search Ads

• When you run a Google search, the ads at the top

are sold by an automated auction.

• Companies can submit a standing bid for a query,
• r can write an agent to update bids over time.
• In each auction, agents submit a single bid, and the

highest K bidders get their ad shown.

• Each winning bidder pays the next-highest bid.
• Over the course of a day, many auctions will be run
• n the same query, or related ones.

Exercise: Model the Ad Auction Game

• Who are the players in the game?
• What decisions do the players have to make?
• What information is available when they make their

decisions?

• What are the sources of a player’s utility?
• What random factors influence the outcome?

I don’t expect you to devise a complete extensive- form game for this example. Instead, you should think about how we can model parts of this interaction using the tools we’ve learned this week.