Content map: SMU H3 Game Theory Map

Chapter 9: Uncertainty and Information

Games of Incomplete Information

Nature and Information

Definition:

Nature selects game ingredients at random and non-strategically.

Players have common knowledge of the game structure but observe only some of Nature’s moves, creating imperfect information.

Strategies and Beliefs

Strategies depend on information about Nature’s move.

Result:

The equilibrium concept stays the same, but players must form beliefs about other players’ information, and these beliefs must be consistent with observed actions whenever possible.

Insight:

Behaviour reveals information because actions depend on information.

Uncertainty and Incomplete Information

Concept

Some uncertainty is external to the players and cannot be controlled directly.
Strategic interaction becomes richer when players hold different information about that uncertainty.

Definition

Definition:

A game of incomplete information is a game in which at least one player is uncertain about a payoff-relevant characteristic, such as types, values, or costs.

External Uncertainty

Definition:

External uncertainty is payoff-relevant uncertainty caused by events outside the decision maker’s control.

Method / Reasoning

Identify what is uncertain.
Identify who knows that information.
Track how actions and messages reveal information.

Implications

Actions can serve both as choices and as signals.
Equilibrium depends on beliefs as well as actions.

Result:

Incomplete information requires analysing strategies and belief updating together.

Insight:

The key question is not only what players want to do, but also what others infer from what they do.

Concept

External uncertainty comes from random events beyond the control of a single decision maker.
The decision maker takes those events as given and chooses an action to maximise expected payoff.

Risk Aversion

Definition:

A risk-averse agent prefers less uncertainty when the average payoff is unchanged.

With Bernoulli utility $u(x)$ , risk aversion is represented by an increasing and concave utility function:

u'(x)>0, \qquad u''(x)<0

For any risky payoff $X$ ,

E[u(X)]<u(E[X])

The agent may prefer a certain payoff to a risky payoff with the same expected monetary value.
Example: if $u(x)=\sqrt{x}$ and $X\in\{40000,160000\}$ with equal probabilities, then

E[X]=100000, \qquad E[u(X)]=300, \qquad u(E[X])=100\sqrt{10}>300

The certainty equivalent is $CE=90000$ because $\sqrt{90000}=300$ .
The risk premium is $100000-90000=10000$ .

Diagram (Bernoulli Utility)

diagram

Insight:

The stronger the dislike of uncertainty, the more the agent values protection against bad states.

Insight:

Risk sharing converts state-dependent payoffs into a more predictable payoff stream.

Risk sharing is useful when outcomes move in opposite directions across agents.
If one agent has a good outcome when another has a bad outcome, transfers can smooth both agents’ payoffs.
A sharing agreement makes the lucky agent compensate the unlucky agent.
The aim is a more stable outcome without changing the average payoff.

Insurance

Insurance is a risk-sharing arrangement.
The insured agent gives up some payoff in good states to receive protection in bad states.
Insurance is valuable because it reallocates payoffs across states.

Result:

Insurance does not create extra average income; it changes who bears risk and when.

Cheap Talk

Concept

Communication can affect equilibrium even when messages are costless and non-binding.
Whether it works depends on how aligned players’ interests are.

Definition

Definition:

Cheap talk is costless pre-play communication that does not directly change payoffs or feasible actions.

Method / Reasoning

Add a communication stage before the underlying game.
Check what happens if players believe and follow the message.
Check what happens if players ignore the message.
See which outcome each player prefers.

Implications

Cheap talk can coordinate players in common-interest environments.
Cheap talk fails in zero-sum settings because messages are not credible.
Bubbling equilibria remain possible: communication may be ignored in equilibrium.

Result:

Cheap talk is informative only when incentives are aligned enough for truthful communication to be optimal.

Insight:

Words matter in games only when the incentives behind the words make them believable.

Adverse Selection

Concept

Private information on quality can destroy trade.
Acceptance or refusal of an offer reveals information to the other side.

Definition

Definition:

Adverse selection is a market failure caused by hidden information, where the uninformed side cannot distinguish high-quality from low-quality types before trade.

Method / Reasoning

Ask which types accept any given offer.
Update beliefs after acceptance.
Check whether the uninformed side still wants to trade at that price.

Implications

Trade may unravel even when mutually beneficial exchanges exist under full information.
Low-quality types remain, while high-quality types exit.

Result:

Markets with hidden information can collapse because the act of accepting an offer changes what the offer means.

Insight:

Selection effects matter because prices influence who stays in the market.

Signalling and Incentive Compatibility

Concept

Informed players may take costly actions to reveal their type.
A type is a player’s private payoff-relevant characteristic (i.e. a characteristic that affects the payoff).

Definition

Definition:

A signalling equilibrium commonly has three possible patterns:

Separating when different types choose different actions.

Pooling when different types choose the same action.

Semi-separating when at least one type randomises, so actions reveal partial but not perfect information.

Incentive compatibility ensures each type prefers the action intended for that type.

Result:

A signal works only if low types do not want to imitate high types.

Incentive Compatibility Inequalities

Let $H$ and $L$ denote the high and low types.
Let $\pi_{a,b}$ denote the payoff of a player with true type $a$ but sends signal $b$ .

Separating Equilibrium

Definition:

For a separating equilibrium where $H$ sends $H$ and $L$ sends $L$ , incentive compatibility requires:

\pi_{H,H} \geq \pi_{H,L}

\pi_{L,L} \geq \pi_{L,H}

Each type must prefer its own signal to imitating the other type.

Pooling Equilibrium

Definition:

For a pooling equilibrium on signal $H$ , incentive compatibility requires:

\pi_{H,H} \geq \pi_{H,L}

\pi_{L,H} \geq \pi_{L,L}

Both types must prefer sending the high-type signal.

Definition:

For a pooling equilibrium on signal $L$ , incentive compatibility requires:

\pi_{H,L} \geq \pi_{H,H}

\pi_{L,L} \geq \pi_{L,H}

Both types must prefer sending the low-type signal.

Semi-Separating Equilibrium

Definition:

For a semi-separating equilibrium, at least one type mixes between signals.

If type $a$ mixes between signals $H$ and $L$ , incentive compatibility requires indifference:

\pi_{a,H} = \pi_{a,L}

If type $b$ uses a pure signal, incentive compatibility requires weak preference for that signal.

Let $q_H$ be the probability that type $H$ sends signal $H$ .
Let $q_L$ be the probability that type $L$ sends signal $H$ .
Receiver beliefs follow Bayes’ rule whenever a signal is observed with positive probability.
Signals are partially informative when $q_H \neq q_L$ but neither signal perfectly identifies type.

Insight:

Semi-separating equilibrium is between pooling and separating: the signal changes beliefs, but it does not fully reveal the sender’s type.

Worked Example: Finding a Semi-Separating Equilibrium

Definition:

Nature draws the sender’s type:

P(H)=\frac{1}{3}, \qquad P(L)=\frac{2}{3}

Sender chooses $S$ or $N$ ; receiver observes the signal, then Accepts or Rejects.

Receiver payoff from Accept is $1$ against $H$ and $-1$ against $L$ ; Reject gives $0$ .

Accepted sender gross payoff is $3$ for $H$ and $1$ for $L$ .

Signal cost is $c_H=0$ , $c_L=\frac{1}{2}$ for $S$ , and $0$ for $N$ .

Sender payoff is gross payoff minus signal cost; Reject gives gross payoff $0$ .

Game Tree

diagram

Look for a semi-separating equilibrium where:

$H$ always sends $S$ .
$L$ sends $S$ with probability $q$ and $N$ with probability $1-q$ .
The receiver accepts after $S$ with probability $r$ ; $N$ is shown as the direct rejection payoff.

Step 1: Find the posterior after signal $S$

By Bayes’ rule,

P(H \mid S) = \frac{P(S \mid H)P(H)} {P(S \mid H)P(H)+P(S \mid L)P(L)}

Since $P(S\mid H)=1$ and $P(S\mid L)=q$ ,

P(H \mid S) = \frac{1\cdot \frac{1}{3}} {1\cdot \frac{1}{3}+q\cdot \frac{2}{3}} = \frac{1}{1+2q}

The signal is partially informative: observing $S$ raises the probability of $H$ above the prior, but does not identify the type perfectly when $q>0$ .

Step 2: Impose receiver indifference

The receiver mixes after $S$ only if Accept and Reject give the same expected payoff.
Reject gives $0$ .
Accept gives

P(H\mid S)(1)+(1-P(H\mid S))(-1) = 2P(H\mid S)-1

Indifference requires

2P(H\mid S)-1=0

P(H\mid S)=\frac{1}{2}

Using the posterior formula,

\frac{1}{1+2q}=\frac{1}{2}

q=\frac{1}{2}

Why impose receiver indifference?

In a mixed strategy, every action used with positive probability must be optimal.
If Accept were strictly better, the receiver would accept with probability $1$ .
If Reject were strictly better, the receiver would reject with probability $1$ .
Therefore, for the receiver to mix after $S$ , the sender’s mixing probability $q$ must make the receiver indifferent.

Step 3: Impose low-type sender indifference

Type $L$ mixes between $S$ and $N$ , so both must give the same expected payoff.
If $L$ sends $N$ , the receiver rejects, so

\pi_L(N)=0

If $L$ sends $S$ , the receiver accepts with probability $r$ .
The signalling cost is paid whenever $S$ is sent, so

\pi_L(S)=r(1)-\frac{1}{2}

Indifference requires

r-\frac{1}{2}=0

r=\frac{1}{2}

Why impose sender indifference?

Type $L$ sends both $S$ and $N$ with positive probability.
If $S$ gave strictly more than $N$ , type $L$ would always send $S$ .
If $N$ gave strictly more than $S$ , type $L$ would never send $S$ .
Therefore, the receiver’s acceptance probability $r$ must make type $L$ indifferent.

Step 4: Check the pure actions

Type $H$ prefers $S$ to $N$ because $S$ gives $r(3)=\frac{3}{2}$ , while $N$ gives $0$ .
After $N$ , the receiver knows the sender is $L$ , so Accept gives $-1$ and Reject gives $0$ .
Therefore, rejecting after $N$ is optimal.

Result:

In this semi-separating equilibrium:

**Sender: **

H: S, \qquad L: S \text{ with probability } \frac{1}{2},

**Receiver: **

\text{Receiver: Accept after } S \text{ with probability } \frac{1}{2}, \quad \text{Reject after } N

The posterior after observing $S$ is

P(H\mid S)=\frac{1}{2}

The signal is informative but imperfect because both types can send $S$ , while only $L$ sends $N$ .

Method / Reasoning

Write each type’s payoff from signalling and from not signalling.
Impose incentive compatibility inequalities.
For any mixing type, impose equality between the signals used with positive probability.
Impose individual rationality inequalities when relevant.

Implications

Productive-looking actions may be chosen partly because they reveal hidden ability.

Insight:

Signals can be socially wasteful even when privately optimal. Credibility comes from the difference in cost, not from the message itself.

Screening and Self-Selection

Concept

Screening reverses the information problem.
The uninformed side designs options to learn information that is relevant to them.

Definition

Definition:

Screening is a mechanism in which process where an uninformed player (principal) acts first to compel an informed player (agent) to reveal private information.

Method / Reasoning

Specify each type’s willingness to pay for each option.
Choose product attributes or contract terms that types value differently.
Choose prices to satisfy incentive compatibility and participation.
Leave just enough surplus for each type to choose the intended option.
Compare menu revenue with uniform-pricing revenue.

Definition:

A menu is incentive compatible when every type weakly prefers its assigned option to all other options in the menu.

For types $H$ and $L$ with assigned options $M_H$ and $M_L$ :

U_H(M_H)\geq U_H(M_L)

U_L(M_L)\geq U_L(M_H)

Menu design is the standard tool: offer several contracts, versions, or price-quality bundles.
Each option must be attractive to its target type and unattractive to the wrong type.
The menu uses self-selection instead of direct identification.

Implications

Firms can increase revenue by offering differentiated versions of the same product.
Successful screening requires each type to prefer its designated option.

Result:

Price discrimination through screening can dominate uniform pricing when types value product attributes differently.

Insight:

The menu’s purpose is to sort types using their own incentives.

Game Theory Chapter 9: Uncertainty and Information

Chapter 9: Uncertainty and Information

Games of Incomplete Information

Nature and Information

Strategies and Beliefs

Uncertainty and Incomplete Information

Concept

Definition

External Uncertainty

Method / Reasoning

Implications

Risk Aversion, Risk Sharing, and Insurance

Concept

Risk Aversion

Diagram (Bernoulli Utility)

Risk Sharing

Insurance

Cheap Talk

Concept

Definition

Method / Reasoning

Implications

Adverse Selection

Concept

Definition

Method / Reasoning

Implications

Signalling and Incentive Compatibility

Concept

Definition

Incentive Compatibility Inequalities

Separating Equilibrium

Pooling Equilibrium

Semi-Separating Equilibrium

Worked Example: Finding a Semi-Separating Equilibrium

Game Tree

Step 1: Find the posterior after signal SSS

Step 2: Impose receiver indifference

Why impose receiver indifference?

Step 3: Impose low-type sender indifference

Why impose sender indifference?

Step 4: Check the pure actions

Method / Reasoning

Implications

Screening and Self-Selection

Concept

Definition

Method / Reasoning

Menu Design Logic

Implications

Step 1: Find the posterior after signal $S$