SMU H3 Map

Content map: SMU H3 Game Theory Map

Sub-Games and Sub-Game Perfect Equilibria

Definition of Sub-Game

Part of the game tree that starts at a single node and that can be considered as a Game in isolation
This means that it can be computed separately using backward induction
This helps calculating the Nash equilibrium when the game reaches the state of a sub-game
The individual decision problem can be replaced with the Nash equilibrium outcome, as it is anticipated that the players will play rationally

Definition of Sub-Game Perfect Equilibrium

Replaces the rollback equilibrium
A pair of strategies, one per player, that constitute an equilibrium in all sub-games
This means every set of strategies which results in a Nash equilibrium in every subgame
This can be found using backward induction

Entry and Pricing Game

Setup

When a circle is made around two choices, it means the player is unaware of the decisions made by the other player
This allows simultaneous games to be mapped out as game trees

Identifying Sub-games

Sub-games can be obtained when games are broken down into their component branches
This simplifies the games as each game can be considered as being isolated
In this game, there are 4 subgames:
- Subgame 1: $(Invest, Invest )$
- Subgame 2: $(Invest, Don't)$
- Subgame 3: $(Don't, Invest)$

Breaking down subgames

Subgame 1

P2 \ P1	High P	Low P
High P	2, 2	-10, 6
Low P	6, -10	-2, -2

For P1, the option of High P is dominated by Low P
For P2, the option of High P is dominated by Low P
There is a weakly dominant strategy of Low P for both players, even if the Pareto efficient strategy is High P

Subgame 2

P2 \ P1	High P	Low P
High P	2, 2	-10, 6
Low P	6, -10	-2, -2

For P1, the option of High P is dominated by Low P
For P2, the option of High P is dominated by Low P
There is a weakly dominant strategy of Low P for both players, even if the Pareto efficient strategy is High P

Subgame 3

P2 \ P1	High P	Low P
High P	2, 2	-10, 6
Low P	6, -10	-2, -2

How to write strategies in each subgame

P1 chooses in $SG_1$ and $SG_2$
P2 chooses in $SG_1$ and $SG_3$
The subgame perfect equilibria are:

\bigg(\{ \text{invest}, (\overbrace{\text{low P}}^{SG_1}, \overbrace{\text{high P}}^{SG_2}) \}, \quad \{ \text{don't}, (\overbrace{\text{low P}}^{SG_1}, \overbrace{\text{high P}}^{SG_3}) \}\bigg)

\bigg(\{ \text{don't}, (\overbrace{\text{low P}}^{SG_1}, \overbrace{\text{high P}}^{SG_2}) \}, \quad \{ \text{invest}, (\overbrace{\text{low P}}^{SG_1}, \overbrace{\text{high P}}^{SG_3}) \}\bigg)

Even if $SG_3$ is never reached if players play rationally, it is still necessary since possibilities for each subgame are needed to represent

Public Good Provision

Setup (First Setup)

Simultaneous decision making
There are 3 players
Each player decides whether to provide or not to provide the public good
If at least 2 players choose to provide the public good, the public good is provided

Player Contributes \ Public Good Provided	No	Yes
No	2	4
Yes	1	3

Subgame Perfect Equilibrium

P1 has a choice at node 0
P2 has a choice at node 1 and node 2
P3 has a choice at node 4, node 5, node 6, node 7
This is represented by:

\{ N, (N, Y), (N, Y, Y, N) \}

The payoffs are $(4, 3, 3)$

In-Out Game

Setup

There are two players
If P1 chooses out, the payoff is $(2, 0)$
If P1 chooses in, P1 and P2 play a simultaneous choice game

P1 \ P2	U	D
U	3, 1	-1, -1
D	-1, -1	1, 3

Subgame Perfect Equilibria

Case 1:

P1 chooses IN
Players pick $(U, U)$
Payoff is $(3, 1)$
For P1, payoff of 3 > 2
Sub-Game Perfect equilibrium of $((\text{IN}, \text{U}), \text{U})$

Case 2:

P1 chooses IN
Players pick $(D, D)$
Payoff is $(1, 3)$
For P1, payoff of 1 < 2
Sub-Game Perfect equilibrium of $((\text{OUT}, \text{D}), \text{D})$

Feds vs. Congress Game

Setup

2 Players, Feds and Congress
Feds (P2) pick interest rate (monetary policy)
Congress (P1) picks budget spending (fiscal policy)

Congress \ Feds	Low IR	High IR
Budget Balance	3, 4	1, 3
Budget Deficit	4, 1	2, 2

Analysis (original)

For P1, Budget Deficit dominates Budget Balance
For P2,
- If P1 chooses Budget Balance, Low IR gives a higher payoff
- If P1 chooses Budget Deficit, High IR gives a higher payoff
The Nash Equilibrium is (Budget Deficit, High IR)

New Rules

Changing order of play:
- From simultaneous game to sequential game
- Congress chooses first
- Feds set the interest rates afterwards

Analysis (new rules)

Let $SG_0$ , $SG_1$ and $SG_2$ be the three subgames

Sub-Game 1 (Budget Balance)

The Nash Equilibrium is $(Balance, Low)$
This results in a payoff of $(3, 4)$

Sub-Game 2 (Budget Deficit)

The Nash Equilibrium is $(Deficit, High)$
This results in a payoff of $(2, 2)$

Sub-Game 0 (Starting Decision)

The Nash Equilibrium is $(Balance, Low)$
This results in a payoff of $(3, 4)$

Sub-Game Perfect Equilibrium

The Sub-Game Perfect Equilibrium is

\bigg( \text{Budget Balance}, (\text{Low}, \text{High}) \bigg)

Alternative Nash Equilibria

A candidate strategy is

\bigg( \text{Budget Deficit}, (\text{High}, \text{High}) \bigg)

If P1 chooses budget balance, P2 chooses high interest rates giving a payoff of $(1, 3)$
Knowing this, P1 chooses budget deficit, making P2 choose high interest rates giving a payoff of $(2, 2)$
If this is the strategy decided, it is a Nash equilibrium since no players are incentivised to change their strategy

Full Payoff Matrix

P1 \ P2	LR, LR	LR, HR	HR, LR	HR, HR
BU BA	3, 4	3, 4	1, 3	1, 3
BU DE	4, 1	2, 2	4, 1	2, 2

This allows us to obtain the Nash equilibria:

\bigg( \text{Budget Deficit}, (\text{High}, \text{High}) \bigg)

This is a Nash equilibrium but NOT the subgame perfect equilibrium strategy

\bigg( \text{Budget Balance}, (\text{Low}, \text{High}) \bigg)

Pastaland vs. Superpizza

Setup

Let Pastaland be P1 and Superpizza be P2
Pastaland and Superpiza must choose independently /simultaneously whether to invest in tech
- $MC_\text{invest} = 3$
- $MC_\text{no invest} = 6$
The cost of investing is $FC_\text{invest} = 4$
After observing both restaurants’ choices, each decides independently and simultaneously on the price of their items
The demand functions are as follows:

q_1 = 21 - p_1 + {p_2 \over 2} \\ q_2 = 21 - p_2 + {p_1 \over 2}

P1 and P2 care about their profits

Perform backward induction to determine whether firms will invest in the technology

Start

There are 4 sub-games
For each sub-games, we need to compute the Nash equilibrium prices and their associated profits
Remember that profits are

\pi = p \cdot q - q \cdot MC = q(p - MC)

Subgame 1 (Don’t, Don’t)

\pi_1 = (p_1 - 6)(21 - p_1 + {p_2 \over 2})

\pi_1 = (p_2 - 6)(21 - p_2 + {p_1 \over 2})

To find the Nash equilbrium, we need to (1) calculate the best response of each firm and (2) solve the system of equations

\text{FOC}_1: 21 - p_1 + {p_2 \over 6} - (p_1 - 6)= 0 \\ 26 - 2p_1 + {p_2 \over 2} = 0

\text{FOC}_2: 21 - p_2 + {p_1 \over 6} - (p_2 - 6) = 0 \\ 26 - 2p_2 + {p_2 \over 2} = 0

\text{at NE, } p_1 = p_2 : \\ 27 - 2p + {p \over 2} = 0 \\ p = 18 \pi = (18 - 6)(21 - 18 + 9) = 12^2 = 144

Subgame 4 (Invest, Invest)

\pi_1 = (p_1 - 3)(21 - p_1 + {p_2 \over 2}) - 4

\pi_1 = (p_2 - 3)(21 - p_2 + {p_1 \over 2}) - 4

To find the Nash equilbrium, we need to (1) calculate the best response of each firm and (2) solve the system of equations

\text{FOC}_1: 21 - p_1 + {p_2 \over 6} - (p_1 - 2)= 0 \\ 26 - 2p_1 + {p_2 \over 2} = 0

\text{FOC}_2: 21 - p_2 + {p_1 \over 6} - (p_2 - 2) = 0 \\ 26 - 2p_2 + {p_2 \over 2} = 0

\text{at NE, } p_1 = p_2 : \\ 27 - 2p + {p \over 2} = 0 \\ p = 18 \pi = (18 - 6)(21 - 18 + 9) = 12^2 = 144

H3 Game Theory Session 5

SMU H3 Map

Sub-Games and Sub-Game Perfect Equilibria

Definition of Sub-Game

Definition of Sub-Game Perfect Equilibrium

Entry and Pricing Game

Setup

Identifying Sub-games

Breaking down subgames

Subgame 1

Subgame 2

Subgame 3

How to write strategies in each subgame

Public Good Provision

Setup (First Setup)

Subgame Perfect Equilibrium

In-Out Game

Setup

Subgame Perfect Equilibria

Case 1:

Case 2:

Feds vs. Congress Game

Setup

Analysis (original)

New Rules

Analysis (new rules)

Sub-Game 1 (Budget Balance)

Sub-Game 2 (Budget Deficit)

Sub-Game 0 (Starting Decision)

Sub-Game Perfect Equilibrium

Alternative Nash Equilibria

Full Payoff Matrix

Pastaland vs. Superpizza

Setup

Perform backward induction to determine whether firms will invest in the technology

Start

Subgame 1 (Don’t, Don’t)

Subgame 4 (Invest, Invest)

Write the subgame perect equilibrium so obtained