Forward Induction

ECON2112

Forward InductionBattle of the Sexes with Entry

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ The game has 3 Subgame perfect equilibria:◮ (InT ,L)◮ (OutB,R)◮ ( 3

4 OutT + 14 OutB, 1

4 L+ 34 R)

◮ Are all of them reasonable?

Forward InductionBattle of the Sexes with Entry

Let us analyze (OutB,R).

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ Player 1 does not move In because is he does, the strategycombination (B,R) will be played in the battle of the sexes.

◮ The strategy combination (B,R) is a rational description ofbehavior in the battle of the sexes.

◮ However, in the context of the current game, is (B,R) a gooddescription of rational behavior in the battle of the sexessubgame?

Forward InductionBattle of the Sexes with Entry

Suppose that we propose the players to play according to (OutB,R).

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ Suppose that player 1 deviates and plays In.

◮ Player 2 was not supposed to move under (OutB,R).◮ How is player 2 going to react when he is called to play?

◮ Will he stick to R?

Forward InductionInformal Definition

Definition (Forward Induction (Informal Idea))Players should make their choices in a way consistent with deductionsabout other players’ rational behavior in the past.

Remember:

Definition (Backwards Induction (Informal Idea))Players should make their choices in a way consistent with deductionsabout other players’ rational behavior in the future.

Forward InductionBattle of the Sexes with Entry

The equilibrium proposed is (OutB,R).

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ Suppose that player 2 is not supposed to move, but he is called tomove anyway. He must, therefore, conclude that player 1 hasdeviated.

◮ When thinking about what to do (whether or not to stick to R) hehas to bear in mind that player 1 is rational.

◮ What will player 2 do?

Forward InductionBattle of the Sexes with Entry

When player 2 has to move in the subgame he has effectively receivedthe following message from player 1:

“Look, I had the opportunity to get 2 for sure, andnevertheless I decided to play in this subgame, and mymove is already made. And we both know that you cannottalk to me because we are in the game, and my move ismade. So think now well, and make your decision.”

Forward InductionBattle of the Sexes with Entry

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ A subgame is a part of the game that could be considered as aseparate game.

◮ However, it should not be treated as a separate game, because ithas been preceded by a very specific way of communication: theplay leading to the subgame.

Forward InductionBattle of the Sexes with Entry

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ If player 2 has to move, he must conclude that player 1 gave up apayoff equal to 2 in an effort to get 4.

◮ By forward induction, player 2 in fact knows that player 1 hasmoved InT .

◮ Player 2 will, consequently, play T .

◮ By backwards induction, player 1 will foresee this and will playInT .

Forward InductionBattle of the Sexes with Entry

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ Set of Subgame Perfect equilibria:◮ {(InT ,L),(OutB,R),( 3

4 OutT + 14 OutB, 1

4 L+ 34 R)}.

◮ Set of Subgame Perfect equilibria conforming with forwardinduction:

◮ {(InT ,L)}.

Forward InductionBurning Money

1. Player 1 decides whether or not burning 2 Dollars.

2. Player 2 observes if player 1 has burnt the money.

3. Players play:L R

T 4,1 0,0B 0,0 1,4

Forward InductionBurning Money

Not BurnBurn

1

B1T1

1

R1

−2,0

L1

2,1

R1

−1,4

L1

−2,0

2

B2T2

1

R2

0,0

L2

4,1

R2

1,4

L2

0,0

2

Forward InductionBurning Money

Not Burn

L2 R2

T2 4,1 0,0B2 0,0 1,4

Burn

L1 R1

T1 2,1 −2,0B1 −2,0 −1,4

1

◮ The game has 9 subgame perfect equilibria.

◮ Only one of them conforms with forward induction.

Forward InductionBurning Money

Not Burn

L2 R2

T2 4,1 0,0B2 0,0 1,4

Burn

L1 R1

T1 2,1 −2,0B1 −2,0 −1,4

1

◮ Suppose player 1 burns the money.◮ By forward induction, player 2 must conclude that player 1 is

playing T1 in the subgame.◮ Note that Burning the money and playing B1 is strictly dominated

by not burning.

◮ If player 1 burns the money player 2 will, therefore, play L1.

Forward InductionBurning Money

Not Burn

L2 R2

T2 4,1 0,0B2 0,0 1,4

Burn

L1 R1

T1 2,1 −2,0B1 −2,0 −1,4

1

◮ By backwards induction, Player 1 knows that he can guaranteehimself a payoff equal to 2 by burning the money.

◮ Suppose now that player 2 observes that player 1 did not burn themoney.

◮ Player 2 knows that player 1 gave up a payoff equal to 2.

◮ Therefore, by forward induction, player 2 must conclude thatplayer 1 is playing T2 in the subgame.

Forward InductionBurning Money

Not Burn

L2 R2

T2 4,1 0,0B2 0,0 1,4

Burn

L1 R1

T1 2,1 −2,0B1 −2,0 −1,4

1

◮ By backwards induction, Player 1 knows that:◮ If he burns the money, player 2 will play T1 in the subgame.◮ If he does not burn the money, player 2 will play T2 in the

subgame.

◮ Consequently, player 1 will not burn the money at the root of thegame.

Forward InductionBurning Money

Not Burn

L2 R2

T2 4,1 0,0B2 0,0 1,4

Burn

L1 R1

T1 2,1 −2,0B1 −2,0 −1,4

1

◮ The unique subgame perfect equilibrium that conforms withforward induction is (Not BurnT1T2 ,L1L2).

◮ (Not BurnT1T2 ,L1L2) gives player 1 a payoff equal to 4 andplayer 2 a payoff equal to 1.

◮ (Having the opportunity to hurt oneself can, sometimes,guarantee a higher payoff.)

SPE and Forward Induction in the Beer-Quiche Game

The Beer-Quiche Game

How to determine whether a SPE equilibrium of such a game satisfies the Forward Induction criterium?

Here’s a three step procedure:

1. Consider an equilibrium

2. For any out-of-equilibrium message m, determine the set S(m) of Player 1’s types that yield a payoff strictly less than the equilibrium payoff whatever Player 2’s best response.

3. Player 2 can attribute a message m only to player 1’s types that do not belong to S(m). If there exists a message m such that, for any best reply of Player 2, there exists at least one type of Player 1 which would deviate to m, then the equilibrium considered does not satisfy the forward induction criterium.

How to determine whether a Nash equilibrium of such a game satisfies the Forward Induction criterium?

Application to the Beer-Quiche game (which as two Nash equilibria in pure strategies: (beersbeerw,notbduelq) and (quichesquichew, duelbnotq).

Step 1: Consider first (beersbeerw,notbduelq) Step 2: m = quiche, S(m) = {surly, wimp} If m = quiche and player 1 is “surly”, then Player 2 best replies by choosing notq

(this will yield him a payoff of 1>0). With such a best reply from Player 2, Player 1 earns 2 which is less than what he would earn in the equilibrium considered in Step 1. Therefore Player 1’s type “surly” belongs to S(m).

If m = quiche and player 1 is “wimp”, then Player 2 best replies by choosing duelq (this will yield him a payoff of 1>0). With such a best reply, Player 1 earns 1 which is less than what he would earn in the equilibrium considered in Step 1. Therefore Player 1’s type “wimp” also belongs to S(m).

Step 3: Since S(m) contains all the possible types of Player 1, Player 1 has no

type that could yield him a higher payoff given Player 2’s best replies. Therefore, the equilibrium considered in Step 1 satisfies the Forward Induction criterium.

How to determine whether a Nash equilibrium of such a game satisfies the Forward Induction criterium?

Application to the Beer-Quiche game which as two Nash equilibria in pure strategies: (beersbeerw,notbduelq) and (quichesquichew, duelbnotq).

Step 1: Consider now (quichesquichew, duelbnotq) Step 2: m = beer, S(m) = {wimp} If m = beer and player 1 is “surly”, then Player 2 best replies with notb (this will

yield him a payoff of 1>0). With such a best reply, Player 1 earns 3 which is more than what he would earn in the equilibrium considered in Step 1. Therefore Player 1’s type “surly” does not belong to S(m).

If m = beer and player 1 is “wimp”, then Player 2 best replies with duelb (this will yield him a payoff of 1>0). With such a best reply, Player 1 earns 0 which is less than what he would earn in the equilibrium considered in Step 1. Therefore Player 1’s type “wimp” belongs to S(m).

Step 3: Since “surly” does not belong to S(m) and since Player 1 earns a payoff of

3 given Player 2’s best reply, Player 1 will deviate from having a quiche upon being “surly”: he will choose to have a beer. Therefore, the equilibrium considered in Step 1 does not satisfy the Forward Induction criterium.