Extensive Form Games 2/10/17
Alpha-Beta Pruning Exercise + − − − + + + + + + − − − − − − − − − 5 6 7 4 5 3 6 8 6 9 4 7 5 6 9 9 8 6 2
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 often 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 other accepts or rejects that division. • Here, we model the agents as getting utility +1 for each unit of resource, but they’re also spiteful so they feel a disutility of 0.5 if they receive less than half. • MANY variations on this model are possible. 1 3,0 0,3 2,1 1,2 2 2 2 2 A R A R A R A R 3,-.5 0,0 2,.5 0,0 .5,2 0,0 -.5,3 0,0
What can’t we model so far? Random Outcomes Incomplete Information Simultaneous Moves
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 0.4,43.6 1 -9,57 2 N 0.4,43.6 .4 .6 -9,57 1 12,-42 1 -11,73 2 8,24 2 -76,-12 -9,57 -31,53 12,-42 29,-30 -11,73 31,3 8,24 Compute expected values: -11*.4 + 8*.6 = 0.4 73*.4 + 24*.6 = 43.6
Incomplete Information 1 2 N .8 .2 1 1 2 2 56,91 68,54 67,49 58,73 39,25 23,33 69,1 20,7 Information Set: A set of decision nodes among which a player can’t distinguish.
Simultaneous Moves 2 1 R P S R S P R 0,0 -1,1 1,-1 1 2 2 2 P 1,-1 0,0 -1,1 R S R S R P S P P S -1,1 1,-1 0,0 0,0 -1,1 1,-1 -1,1 0,0 1,-1 1,-1 -1,1 0,0 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, or 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 on 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.
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