1. Computing Approximate Bayes-Nash Equilibria in Tree-Games of Incomplete Information
Satinder Singh, Vishal Soni, Michael P. Wellman University of Michigan Computer Science and Engineering Published In Proceedings of the Fifth ACM Converence on Electronic Commerce (EC), pp , Presented By Brian Kirby

2. What Is Game Theory? Uncertainty Revisited Other Agents' Decisions
Minimax Search (Turns)
Not Always “Fun” Games

3. Game Components
Players
Actions
Payoff Matrix

4. CS5811 Student's Dilemma
Dr. Fonder just began teaching the CS5811 course at a public university. Unfortunately, she has only two students: Rolando and Christophe.
At the end of the semester, Dr. Fonder writes a particularly difficult take-home exam for the class.
In class Rolando and Christophe both notice that the fifth question is especially difficult, and indicate this to Dr. Fonder

5. CS5811 Student's Dilemma Dr. Fonder makes the following deal:
“At least try question #5. See if you can do it. If both of you me to ask for help, I was unfair, so I will omit the question so you don't lose points. But, if only one of you asks for help and the other doesn't, I will omit the question for the student that DIDN'T ask me for help. Further, the student that doesn't ask for help can steal 10 of the other student's points!”

6. Payoff Matrix

7. Basic Terms
Strategy
Rational Strategy
Solution To A Game

8. But It's Not The Best!
Why Not Choose Best?
Nash Equilibrium

9. Two Nash Equilibrium Points

10. Applied To Tree Games Direct Action Dependencies Agent 1 Agent 3

11. Tree-Games With Complete Information
Agent Payoff := function(All Agent's Actions)
Locality Used To Get Nash Equilibrium
Agent 1
Agent 3
Agent 2

12. Problem With Nash Equilibrium
Assumes Equilibrium Strategy
Real World: Many “Types” of Agents
What Is A Type?

13. Example Agent 1 := Worker Agent 2 := Employer
Worker “Type” := skilled | unskilled
Prior Probability Distribution
Adjusted Probability Distribution
Bayes Law Given Actions

14. Bayes-Nash Equilibrium
Before
Agent Payoff := function(All Agent's Actions)
Now
Agent Payoff := function(All Agent's Actions, type)
Choose Actions Without Knowing Other's Type
Probability Distribution For Type (Changes)

15. Back To Tree-Games Goal: Compute Bayes-Nash Equilibrium
Used in games with incomplete information
Other Agent type is not known.
Already Can Compute Nash Equilibrium
Agent 1
type
Agent 3
type
Agent 2
type

16. Problem Types Tree-games with Discrete Types
Tree-games with continuous types
Constraints on effect of type on payoff

17. Basic Idea of Abstract-TreeBNE
Compute Depth-First Ordering For Nodes
Downstream Pass
Ask “w” if x's strategy is good response to y's
If “w” agrees, add “w” to “witness list”
Set D(xStrategy, wStrategy) = 1
Pass entries for w, x strategy back up w x y

18. BNE Basic Algorithm Used For Discrete Types
Continuous w/ constraints on effect on payoff
Only Approximation
Complete Information Algorithm Is Exact

19. Questions?