```Towards Equilibrium Transfer in
Markov Games

2013-9-9
Outline
Background
Preliminary Ideas
Some Results
Background
Multi-agent Reinforcement Learning
Single-agent RL:
Path finding
Mountain Car
Robot Soccer
IKEA furniture robot
Markov Games
: < , , ,  >
: the discrete state space.
: the action space of the agent.
:  ×  →  is the reward function.
:  ×  ×  → 0,1 is the transition function.
from one agent to more
than one
: < , ,  =1… ,  =1… ,  >
N: the set of agents.
: the discrete state space.
= 1 × ⋯ ×  : the joint action space of the
agents.
:  ×  →  is the reward function.
p:  ×  ×  → 0,1 is the transition function.
Agent take joint
actions
Equilibrium-based MARL
Some equilibrium solution concepts in game theory can be adopted
Our Previous Work
 Equilibrium-based MARL:
 Multi-agent reinforcement learning with meta equilibrium []
 Multi-agent reinforcement learning by negotiation with
unshared value functions []
 Focusing on combining MARL with equilibrium solution
concepts
 Problematic issues:
 Equilibrium computing is complicated and time consuming
 A new complexity class: TFNP! []
 For tasks with many agents, equilibrium-based MARL
algorithms may take too much time
How to accelerate the learning process of equilibrium-based MARL?
Transfer Learning in RL
Matthew E Taylor, Peter Stone. Transfer learning for reinforcement learning domains.
Journal of Machine Learning Research, 2009.
Alessandra Lazaric. Transfer in reinforcement learning: a framework and a survey.
Reinforcement Learning, Springer, 2012.

instance/policy/value
function/model/…
′
accelerate
Reuse learnt
knowledge
Transfer Learning in Markov Games?

instance/policy/value
function/model/…
′
……
Why not transfer between
these normal-form games
within a Markov game?
……

 Transfer equilibrium between similar normal-form
games during learning in a Markov game:
 Reuse the computed equilibria in previous games
 Reducing learning time
 Key problems:
 Which games are similar?
 For example: the games occur on different visits of a state
 How to transfer equilibrium?
1+1 (, , )
1 (, , )
(, )
1
2
(, )
1
2
1
2
1
1
2
1
2
−1
0
2
−1
−0.5
……
Preliminary Ideas
Game Similarity
 Games with the same action space?
 Games with different action space?
 Similarity payoff distance?
 Equilibrium-based similarity or equilibrium-independent
similarity?
Drew Fudenberg and David M. Kreps. A theory of learning,
experimentation and equilibrium in games. 1990.
Game Similarity
Find equilibria of two games
and compute the similarity
Equilibrium-based similarity
Weird Cycle
Transfer seems senseless!
Equilibrium transfer
Why not take (, ) in the second game?
Our Idea
Transfer equilibrium between games which are thought to be similar.
Evaluate how much the loss brought by equilibrium transfer is.
Transfer is acceptable when there is a little loss.
1+1 (, , )
1 (, , )
(, )
1
2
(, )
1
2
1
2
1
1
2
1
2
−1
0
2
−1
−0.5
The two games are different only in one item.
……
Problem Definition
(, )
1
2
1
1
2
1
0
2
−1
−0.5
(, )
1
2
1
2
2
−1
, ∗
transfer method?
′, ?
 Can we find a transfer method which can transfer the
computed Nash equilibrium ∗ in game  to a strategy
profile ′ in game ′ that satisfies ∀ ∈  and ∀ ∈  ,
there holds
Approximate
′
′ ′
′
, − ≤   + , Nash equilibrium
where  is close to 0.
 In other words, given a transfer method, if  is small
enough, then the transfer method is acceptable.
 Furthermore,
Problem Definition
 ∀ ∈  and ∀ ∈  , define the transfer error
′
′ ′
′
, ′ =   , − −
 Let  ′ = max  ( , ′ )

 Let  ′ = max  (′ )

Given a transfer method, we need to find the bound of (′ )!
A Naïve Transfer Method
Direct Transfer
1
2
1
1
2
1
0
2
−1
−0.5
1
2
1
2
2
−1
, ∗
∗
(, )
(, )
′, ?
 Define the difference of the two games  =  ′ −  such
that ∀ ∈  and ∀ ∈
=  ′  −   .
 Examine the transfer error
′
′ ∗
∗
′
∗
,  =   ,  =   , − −
A Naïve Transfer Method
′
′
∗
, ′ =   , −
−  ∗
′
′
∗
= Σ− −
−   , − − Σ′ Σ− ∗ ′  (′ , − )

∗
= Σ− −
−
′
′
, − − Σ′ ∗ ′  ′ , −

∗
= Σ− −
− [  , − +  ( , − ) − Σ′ ∗ ′ [ ′ , − +  (′ , − )]]

∗
= Σ− −
−   , − − Σ′ ∗ ′  ′ , −

∗
+ Σ− − − [ ( , − ) − Σ′ ∗ ′  (′ , − )]

∗
≤ Σ− −
− [ ( , − ) − Σ′ ∗ ′  (′ , − )]

∗
= Σ− −
−   , − − Σ ∗   ()
∗
= Σ− −
− +  , − − Σ ∗   ()
≤ Σ− +  , − − Σ ∗   ()
+  , − = max(0,   , − )
A Naïve Transfer Method
Σ− +  , − − Σ ∗   ()
Many items in  are zero if two games are very similar
Some Results
Future Work
 Some problems:
 Other transfer methods?
 Only Nash equilibrium?
 Equilibrium finding algorithms
 Transfer between games with different action space
 Transfer between games with different agent numbers
 Game abstraction
Thanks!
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