How to play when you are not sure?ananthap/images-static/seminarWeek'18.pdfAnantha Padmanabha How to play when you are not sure? Perfect information games Logic for perfect information

Perfect information gamesLogic for perfect information games

Imperfect information gamesLogic for imperfect information games

How to play when you are not sure?

Anantha Padmanabha

Institute of Mathematical Sciences, Chennai

Institute seminar day16 April 2018

Anantha Padmanabha How to play when you are not sure?



1 Perfect information games

2 Logic for perfect information games

3 Imperfect information games

4 Logic for imperfect information games




Game Tree

Example

Player A picks one action from {a1, a2, a3} and player B respondswith one action from {b1, b2}. The label in the leaf denotes whowins.

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2




Strategy tree

Extensive form Game

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

Strategy for player B

A

B1 B2 B3

B A B

a1 a2 a3

b1 b1 b2

Not winning for player B.




Strategy tree

Extensive form Game

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

Strategy for player B

A

B1 B2 B3

B A B

a1 a2 a3

b1 b1 b2

Not winning for player B.




Winning strategy

Extensive form Game

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

Winning strategy for player B

A

B1 B2 B3

B B B

a1 a2 a3

b1 b2 b2




Logic and Games

Logic can be used as a formal language to express some propertiesabout these games and strategies.

Some interesting properties

- w is a leaf node and player B wins at w .

- Tree node w belongs to player A.

- From the tree node w the action a1 is active (can beperformed).

- σ is a winning strategy for player B (σ refers to a subtree).

- There is some leaf which is in the subtree rooted at w whereA wins.

The logic should be able to talk about both games and strategies.




Logic and Games












Logic and Games












Logic and Games












Logic and Games












Logic and Games












Logic and Games












Logic for perfect information games

Ramanujam and Sunil introduced this logic to talk about suchgames and strategies. 1

Syntax- p

Properties about a game nodes. Example:

p1 ::= Player A wins p2 ::= is Player B node.p3 ::= is root p4 ::= is leaf.

- 〈a〉αFrom the current node w there is an a a-action successor vwhere α is true

- 〈a〉αFrom the current node w , the incoming label is a and α istrue at the parent of w .

1where game trees are unravellings of a finite graph.Anantha Padmanabha How to play when you are not sure?





Syntax- p










Syntax- p










Syntax- p









Syntax (Contd.)

- ♦-αFrom the current node w there is some ancestor v where α istrue.

- (σ)i : aIf player i is playing according to σ then she has to play actiona at the current world w .

σ i αIf player i plays according to strategy σ then α is guaranteedto hold.





Syntax (Contd.)








Syntax (Contd.)







Example

Game tree

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

Strategy σ of player B

A

B1 B2 B3

B B B

a1 a2 a3

b1 b1 b2

Some formulas

〈b1〉(winA) True at B2,B3 and false everywhere else

(σB) : b2 True at B3 and false everywhere else.

σ B winB True at all nodes.




Example

Game tree

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2


A

B1 B2 B3

B B B

a1 a2 a3

b1 b1 b2

Some formulas



σ B winB True at all nodes.




Example

Game tree

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2


A

B1 B2 B3

B B B

a1 a2 a3

b1 b1 b2

Some formulas



σ B winB True at all nodes.Anantha Padmanabha How to play when you are not sure?



Suppose player A’s action is private (She whispers it to thereferee),

and then the referee announces first2 if A has chosen a1or a2 and announces last if A has chosen a3.

Imperfect information game tree

If referee announced first2 then there is no way for B to saywhether the current position is B1 or B2.

It means that B1 and B2 are indistinguishable nodes for playerB.

A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

B




Suppose player A’s action is private (She whispers it to thereferee), and then the referee announces first2 if A has chosen a1or a2 and announces last if A has chosen a3.




A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

B








A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

B








A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

B








A

B1 B2 B3

B A A B A B

a1 a2 a3

b1 b2 b1 b2 b1 b2

B




Properties of indistinguishable nodes

If node u and node v have are indistinguishable for someplayer A then the actions available at u should be the same asactions available at v for A.Otherwise A will be able to distinguish the nodes based on theactions that are available to him.

The strategy of player A has to suggest same action at u andv if they are indistinguishable.

The strategy should depend only on what player A knows /able to tell for sure.




Properties of indistinguishable nodes

If node u and node v have are indistinguishable for someplayer A then the actions available at u should be the same asactions available at v for A.Otherwise A will be able to distinguish the nodes based on theactions that are available to him.

The strategy of player A has to suggest same action at u andv if they are indistinguishable.

The strategy should depend only on what player A knows /able to tell for sure.




Can indistinguishable nodes be in different levels?

Absent minded driver problem

A wants to go home from office. But if he takes a wrong turn hewill be eaten by dinosaurs. Unfortunately both junctions lookidentical and to add to it, A is absent minded.

Here the indistinguishable nodes are in two different levels.




Can indistinguishable nodes be in different levels?

Absent minded driver problem

A wants to go home from office. But if he takes a wrong turn hewill be eaten by dinosaurs. Unfortunately both junctions lookidentical and to add to it, A is absent minded.

Here the indistinguishable nodes are in two different levels.




As a first step, we look the logic for only the game part withoutmentioning the strategies (Padmanabha, Ghosh).

Syntax

- p, 〈a〉α, 〈a〉α, ♦-αAll old constructs not involving strategies (σ).

KAαIf the current node is w then α is true in all the nodes thatare indistinguishable from w for player A.

This logic can express some interesting properties in the imperfectgame settings.





Syntax








Syntax







Conclusion

We have introduced logic suitable to model games with imperfectinformation.

To be done: Incorporate imperfect strategies into the logic.




Conclusion

We have introduced logic suitable to model games with imperfectinformation.To be done: Incorporate imperfect strategies into the logic.





Documents

How to play when you are not sure?ananthap/images-static/seminarWeek'18.pdfAnantha Padmanabha How to play when you are not sure? Perfect information games Logic for perfect information