SMU H3 Map

Content map: SMU H3 Game Theory Map

Tit-for-Tat for the Infinite Game

Definition

Strategy copies the behaviour from $t - 1$ for the game at timestep $t$
4 classes of subgames matter
- $(C, C)$
- $(C, D) \rightarrow (D, C)$
- $(D, C) \rightarrow (C, D)$
- $(D, D)$

Payoffs

Subgame 1

V(C, C) = 4 + \delta V(C, C)

V(C, C) = \frac{4}{1 - \delta}

Subgame 2

V(D, D) = a + \delta V(D, D)

V(D, D) = \frac{a}{1 - \delta}

Subgame 3

V(C, D) = 6 + \delta V(D, C)

V(C, D) = 6 + \delta(-2 + \delta V(C, D))

V(C, D) = 6 + -2 \delta + \delta^2 V(C, D)

V(C, D) = \frac{6 - 2 \delta}{1 - \delta^2}

Subgame 4

V(D, C) = -2 + \delta V(C, D)

V(D, C) = -2 + \delta (6 + \delta V(D, C))

V(D, C) = -2 + 6\delta + \delta^2 V(D, C)

V(D, C) = \frac{6\delta - 2}{1 - \delta^2}

Answer

Subgame 1

We are in the subgame after observing $(C, C)$
$V(C, C) = 4 + \delta V(C, C)$ if no deviation
$V(\text{one-shot deviation}) = 6 + v(D, C)$ if deviation
To deviate, we need

V(C, C) \ge 6 + \delta V(D, C)

Subgame 2

Subgame 3

Subgame 4

$V(D, D) = a + \delta V(D, D)$ if no deviation
$V(\text{one-shot deviation}) = -2 + V(C, D)$ if deviation
To deviate, we need

V(D, D) \ge -2 + \delta V(C, D)

Carrot Game

Rules

Two players with one box each
One player looks into the box to see if he has a carrot or not
The other player has to decide whether to swap boxes or keep the box that he was given
The objective of the game is to receive the carrot
There is imperfect information since only one player knows where the winning condition is

External Uncertainty

Definition

Random events that are beyond the control of the single decision maker
Who then takes them as given and acts in order to maximise his or her expected payoff
Agents may not like taking risk (risk aversion) and so they look for solutions to protect themselves against it

If things go well for me, they go bad for you and vice versa

Agreement 1

Outcome	Good	Bad
Results	160,000	40,000
Probability	0.5	0.5

\mathbb E (\pi) = 100000

Agreement 2

If an agreement is made such that the lucky one pays 60000 to the unlucky one

Outcome	Good	Bad
Results	100,000	100,000
Probability	0.5	0.5

Risk sharing occurs when you have a definite outcome of the expected payoff on average without incurring risk

Risk-Aversion

A risk-averse agent minimises uncertainty even when the expected payoff is the same

Formalising Risk-Aversion

Suppose that we value the amount of $x$ at $\sqrt x$
Expected payoff from being payoff-dominant is

\mathbb E(u) = \frac{1}{2} \sqrt{160000} + \frac{1}{2} \sqrt{40000} = 300

Expected payoff from the risk-dominant agreement is

\mathbb E(u) = \frac{1}{2} \sqrt{100000} + \frac{1}{2} \sqrt{100000} = 100\sqrt{10} > 300

Insurance

Definition

Insurance companies take some risk in exchange of cash
Going back to the example, if I had 90,000 in each state, I would be getting the same expected payoff of 300
Since by being subject to risk, on average I have 100,000
I can conclude that 10,000 is the maximum price I am willing to pay (to the insurance company) to forgo risk

Strategic Information Transmission

Definition

The extent to which the informed party wants to reveal the information is a matter of strategic considerations
The informed party should decide to reveal what is in his best interest to reveal
The uninformed party is aware of this fact

Cheap Talk

Definition

The easiest way to transmit information is by communicating it
When this does not cost anything to the player, we say we are in the context of Cheap Talk games
Cheap Talk games add one round of pre-play communication to the game

Insights

If the interests in the game are aligned, pre-play communication may help solving coordination problems
However there are other sub-game perfect equilibria in which the pre-play communication is ignored (babbling equilibria)

Rules

Player 1 announces his choice
Both players maker their choices after the announcement
Cheap Talk works in this case:
- Player 1 says “Local Latte” and then chooses to do whatever he has said at the pre-play communication
- Player 2 chooses to go wherever Player 1 said he was going
Cheap Talk does not work in this case:
- Player 1 says “Local Latte” and then chooses Starbucks no matter his announcement
- Player 2 chooses Starbucks no matter what Player 1 has said

	Starbucks	Local Latte
Starbucks	$1,1$	$0,0$
Local Latte	$0,0$	$2,2$

Zero-Sum Games

Cheap Talk does not work in zero sum games
In other words, when the interests between the two players are in total conflict, cheap talk does not work

Lemon Market

Example 1

If the value of a used car can be either $0$ or $10$
- If the seller offers $0$ , buy since the expected payoff is $0$
- If the seller offers $P > 0$ , do not buy since the seller will sell the car worth $0$ , meaning that the expected payoff is $-P$
The seller who knows the value of the used car accepts an offer to buy that car at the average price $5$

Example 2

Suppose the value of the car is $q$ for the seller
Suppose the buyer is willing to pay $1.5 q$ for the car (but he does not know $q$ )

Illustration

When you pay $10$ , the average value of the car is $\frac{0+10}{2} = 5$
To the buyer, the value of the car is $1.5 \cdot 5 = 7.5$
You lose on average, since $10 > 7.5$

Signalling

Informed parties can credibly convey information to the uninformed party if this is convenient to them
Signalling is a strategy used to resolve asymmetric information, where one player has information that the other player lacks

Education Game

Setup

Student $A$ and Student $C$
They take courses at school:
- Type $A$ can take hard courses at $\$ 3000 $
- Type $C$ can take hard courses at $\$ 14000 $
Employers value workers
- Type $A$ at $\$ 160000 $
- Type $C$ at $\$ 60000 $

Signalling Function

Type $A$ may find it worthwhile signalling their type by taking enough hard courses $n$ that type $C$ students do not want to take
The signal works if it cannot be faked, that is:
- Type $A$ students prefer to take $n$ hard courses (rather than deviating to choose $0$ courses)
- Type $C$ students prefer to take $0$ hard courses (rather than deviating to choose $n$ courses)

Incentive Compatibility

$n$ must satisfy the following inequalities
Upper bound

160,000 - 3,000n \ge 60,000 \\ n \le {100 \over 3}

Lower bound

60,000 \ge 160,000 - 14,000n \\ n \le {100 \over 14}

This means that there is no gains from deviating in strategy, meaning that the student is indifferent

\therefore 7.14 \le n \le 33.33

8 \le n \le 33

Individual / Participation Rationality

Suppose type $A$ students can secure at least $\$ 125,000 $
Suppose type $C$ students can secure at least $\$ 30,000 $
Type $C$ is always incentivised to participate ( $\$ 60000 > $30000 $)
Type $A$ is only incentivised to signal such that

160000 - 3000n \ge 125000

n \le {35 \over 3} \approx 11

Thus, $8 \le n \le 11$

Pooling Equilibrium

A separation may not be the only possible outcome
Pooling of types may occur as well, in which case both $A$ and $C$ students earn the same wage, and none of them take hard course while in college
The common wage is the expected value from hiring an A or a C student, that is

160,000p + 60,000(1-p) = 100,000p + 60,000

While a $C$ student prefers pooling from separation, an $A$ student does not
So this type may have an incentive to deviate and take just one hard course
We need that employers understand that the deviation comes from the $A$ student

160,000 - 3,000 > 100,000p + 60,000

Thus, deviations are not profitable and pooling equilibrium occurs when

160,000 - 3,000 \le 100,000p + 60,000

p \ge 0.97

We need $p$ large enough for pooling equilibrium to arise in this context

H3 Game Theory Session 8

SMU H3 Map

Tit-for-Tat for the Infinite Game

Definition

Payoffs

Subgame 1

Subgame 2

Subgame 3

Subgame 4

Answer

Subgame 1

Subgame 2

Subgame 3

Subgame 4

Carrot Game

Rules

External Uncertainty

Definition

Risk Sharing

Agreement 1

Agreement 2

Risk-Aversion

Formalising Risk-Aversion

Insurance

Definition

Strategic Information Transmission

Definition

Cheap Talk

Definition

Insights

Rules

Zero-Sum Games

Lemon Market

Example 1

Example 2

Illustration

Signalling

Education Game

Setup

Signalling Function

Incentive Compatibility

Individual / Participation Rationality

Pooling Equilibrium