RMM

1
RMM

• Or how to decide taking into account what I think
  that the other thinks that I think that the other
  thinks...

2
Multiagent Rational Decisions

• Without other rational agents
• With other rational agents
• Game Theory equilibria
• Recursive Decision Analysis

3
Utility matrices
4
Example.- Observation Robots
5
Possible actions

• a11 R1 goes to P1
• a12 R1 goes to P2
• a13 R1 does nothing
• a21 R2 goes to P1
• a22 R2 goes to P2
• a23 R2 does nothing

6
Utility matrix calculation
7
Utility matrix calculation
8
What will R2 do?Case 1.- R2 knows about P2
9
What will R2 do?Case 1.- R2 knows about P2
10
Case 2.- R2 does not know about P2
11
Case 2.- R2 does not know about P2
12
Probabilities
P0.9
P0.1
13
Model nesting

• ...but R2 needs to know what R1 thinks in order
  to make a rational decision.
• This could continue indefinitely
• At some point recursive mutual knowledge ends.
• No-info nodes

14
(No Transcript)
15
No-infos

• No-info2 indicates that R1 knows R2 does not have
  knowledge to model R1
• No-info1 represents lack of knowledge in R1 about
  whether it is being modeled by R2 or how.
• No-info could be considered as a uniform
  probability distribution over actions.

16
Solving the decision problem

• We proceed bottom-up
• First we calculate the expected utility of
  subtrees, then the utility of nodes.
• Process ends when root is reached.

17
Leaves calculation

• If R2 does not know about P2, R2 expected
  utilities are
• u(a21) 0.50 0.50 0
• u(a23) 0.52 0.50 1 (wins)
• If R2 could see P2, its utilities are
• u(a21) 0.330 0.334 0.330 1.33
• u(a22) 0.335 0.333 0.333 3.67 (wins)
• u(a23) 0.332 0.334 0.330 2

18
Reconsidering R2

• With probability p 0.9 decides a23 (do
  nothing)
• With probability p 0.1 decides a22 (go to P2)
• So with probability p 0 decides a21 (go to P1)

19
Upwards propagation

• Utilities of R1 actions are
• u(a11) 01 0.15 0.91 1.4
• u(a12) 04 0.12 0.92 2
• u(a13) 02 0.14 0.90 0.4
• We conclude that R1 decides to go to P2
• R1 believes it is so unlikely that R2 goes to P2,
  that it better goes itself.

20
Comparison with Game Theory

• Game Theory equilibria search does not apply
  because there is no common knowledge
• (R1 does not know if R2 knows what R1 knows about
  P2, etc).