Simultaneous-Move Games 4/21/17 Recall: Game Trees 1 3,0 0,3 - - PowerPoint PPT Presentation

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Simultaneous-Move Games 4/21/17 Recall: Game Trees 1 3,0 0,3 - - PowerPoint PPT Presentation

Aug 13, 2023 133 likes •286 views

Simultaneous-Move Games 4/21/17 Recall: Game Trees 1 3,0 0,3 2,1 1,2 2 2 2 2 A R A R A R A R 3,-.5 2,.5 0,0 -.5,3 0,0 0,0 .5,2 0,0 Agents make decisions sequentially. Outcomes (leaves) have a utility for each

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SLIDE 1

Simultaneous-Move Games

4/21/17

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SLIDE 2

Recall: Game Trees

  • Agents make decisions sequentially.
  • Outcomes (leaves) have a utility for each agent.
  • Solve by backward induction.

Good for modeling:

  • Classic board games (connect four, hex, etc.)
  • Discrete time interactions (resource sharing, deterrence)

1 2 2 2 2 3,-.5 0,0 2,.5 0,0 .5,2 0,0

  • .5,3

0,0 3,0 2,1 1,2 0,3 A R A R A R A R

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SLIDE 3

Simultaneous Moves

What if agents make decisions at the same time?

  • Rock-paper-scissors

2 1 R P S R 0,0

  • 1,1

1,-1 P 1,-1 0,0

  • 1,1

S

  • 1,1

1,-1 0,0

  • normal form game
  • payoff matrix
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SLIDE 4

Navigation Example (sequential)

A B

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SLIDE 5

Exercise: Construct a game tree.

Available actions: Up, Right Up, Left Timing: short path: 30s 45s long path: 40s 60s conflict: adds 20 seconds to both

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SLIDE 6

Navigation Example (simultaneous)

A B

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SLIDE 7

Bonus question: what if a conflict only costs 5s?

Exercise: Fill in the payoff matrix.

U L U R

Convention: Row player’s utility first.

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SLIDE 8

Nash Equilibrium

  • An outcome where no agent can gain by unilateral

deviation.

U L U

  • 60,-80
  • 40,-45

R

  • 30,-60
  • 50,-65

U L U

  • 45,-65
  • 40,-45

R

  • 30,-60
  • 35,-50
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SLIDE 9

Identifying Nash Equilibria

for each cell in the payoff matrix: for each player: for each deviation: if deviation > strategy: cell is not a NE if no beneficial deviations: cell is a NE

D E F A

9,3 1,4 7,3

B

4,1 3,3 6,2

C

  • 1,9

2,8 8,-1

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SLIDE 10

Dominated Strategies

If strategy X is always better than strategy Y, then Y can be eliminated from the game. After one strategy is eliminated, others may be dominated in the game that remains.

D E F A

9,3 1,4 7,3

B

4,1 3,3 6,2

C

  • 1,9

2,8 8,-1 Exercise: Iteratively eliminate dominated strategies until no more dominated strategies remain.

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SLIDE 11

Pursuit/Evasion Example

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SLIDE 12

Exercise: Construct a payoff matrix.

Available actions: Left, Right Left, Right Incentives: +5 catch:+1 +3 miss: −1

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SLIDE 13

Exercise: Identify all Nash equilibria.

  • There are no pure-strategy equilibria!
  • We need mixed strategies
  • Agents should deliberately randomize their actions

L R L

0,1 5,-1

R

3,-1 0,1

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SLIDE 14

More than two agents

  • Add more dimensions to the array.
  • Add more utilities to each outcome tuple.

In a game with N agents that each have S actions, how many utilities does the payoff matrix contain?

u1,u2,u3

Player 1 Player 2