Extensive Form Games
• Describing Extensive Forms
• Nash Equilibrium
• Solving extensive forms ``backwards’’
• Subgame Perfection
Nash Equilibrium
• Specification of strategies such that each
player is maximizing his/her payoff given the
strategies of the others
• Each player’s strategy is a best response to
the strategies of the others
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
• Either stop or pass to other player
• If pass to other player, the total size of the pie (sum
of payoffs) doubles
• If stop, get 4/5 of pie and the other player gets 1/5
Centipede Game
• Let us look for pure strategy Nash Equilibria
Centipede Game
• Let us look for pure strategy Nash Equilibria
• Highest payoff would be to continue at each
node
• Is that an equilibrium?
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
• Always pass is not an equilibrium
• What about pass until last move then stop?
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
• What about pass until the second to last
move then stop?
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
256
64
Centipede Game
• In a pure strategy Nash Equilibrium:
• If expect to stop at some node, then should
stop at the previous node
• Only possible outcome: stop at first node!
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
stop
32
128
A Pure strategy Nash equilibrium: Play stop at
every node
256
64
Centipede Game
1
stop
4
1
pass
2
stop
2
8
pass
1
stop
16
4
pass
2
stop
8
32
pass
1
stop
64
16
pass
2
pass
256
64
stop
32
128
All Nash equilibria involve stopping at the first node
(some mixed Nash equilibria involve low probability of
continue at second node, strategies beyond can be
arbitrary in Nash equilibrium)
Quiz Consider this cen5pede game: Time : 1 2 3 4 5 6 1
stop 5 5 pass 2
stop 10 10 pass 1
stop 15 15 pass 2
stop 20 30 pass 1
stop 50 20 pass 2
stop 30 50 pass 40 40 Quiz Which could be a pure-­‐strategy Nash equilibrium outcome: a) Stopping at 5me 1 with (5, 5). b) Passing at 1 and Stopping at 2 with (10, 10). c) Passing at 1 & 2 and Stopping at 3 with (15, 15). d) Passing at nodes 1 through 3 and Stopping at 5me 4 with (20, 30). Quiz Explana.on (d) is true. •  Check that (a) is not an equilibrium outcome: stopping at 1 cannot be a pure strategy Nash equilibrium because all the outcomes aKer it give 1 a strictly higher payoff and so 1 is beNer off passing. (The same argument works for (b) and (c).) •  Passing at 1 through 3 and Stopping at 4 with (20, 30) is a pure strategy Nash equilibrium outcome: –  If 2 an5cipates that 1 will stop at 5, then it is a best response for 2 to stop at 4 and get (20, 30) which is beNer than (50, 20) for player 2. The passing at nodes 1 through 3 are also best responses, as are any ac5ons aKer node 4 since they are never reached.