Lecture 10: Decision Making Under Uncertainty

Lecture 10: Decision Making Under Uncertainty

Ram Singh

Department of Economics

February 10, 2015

Lotteries I

Consider the outcomes/states of nature that follow from tossing of a coin:

Outcomes belong to the set{H,T};

Outcomes may be equiprobable;

Outcomes may not be equiprobable.

If outcomes are equiprobable, we can denote the experiment as a lottery/risky-alternative by

(pH,pT) = (1 2,1

2).

If the coin is biased, it will give us a different lottery say(¹₃,²₃).

Different coins, in principle, will generate different lotteries.

Lotteries II

Examples of lotteries:

Output/profit from a project; Low or High Result of an Exam; Pass or Fail

Outcome of a career-path

Outcome of the Placement Process at DSE; Selected or Not In general let

Ω ={s₁,s₂, ...,s_S} be the set of outcomes. For this setting we define Definition

Simple Lottery: is a vectorL= (p1, ...,pS), such thatps≥0 andP

s(ps) =1.

psis the probability of the occurrence of outcomes.

Lotteries III

LetLbe the set of simple lotteries.

A typical element ofLisLk = (p1, ...,p_S);k^thlottery.

Example Suppose,

L={(p₁,p₂,p₃)|p_i ≥0andX p_i =1}

Simple Lotteries;L1= (1,0,0),L2= (0,1,0),L3= (0,0,1),L4= (¹₂,¹₂,0), L5= (¹₂,¹₆,¹₃)∈L.

Probability Spaces and Lotteries

Consider the experiment of tossing of a coin:

Let,

Ω ={H,T}

ϑ={φ,{H},{T},{H,T}}

µ:ϑ7→[0,1]

(∀A∈ϑ)[0≤µ(A)≤1]

µ(Ω) =1.

µis a measure of (objective) probability over the elements ofϑ, e.g., µ(φ) =0,

µ({H}) = 1

2 =µ({T}), µ({H,T}) =1.

We call(Ω, ϑ, µ)to be a probability space.

Probability Spaces and Lotteries

Consider another experiment: Casting of dice.

Let

Ω ={s₁,s2,s3,s4,s5,s6}

ϑ={φ,{s₁},{s₂}, ...,{s₁,s2,s3,s4,s5,s6}}

µ:ϑ7→[0,1]

(∀A∈ϑ)[0≤µ(A)≤1]

µ(Ω) =1.,e.g., µ(φ) =0, (∀i)µ({s_i}) =1

6.,etc (Ω, ϑ, µ)is another probability space.

Lottery Types I

Consider three lotteries:L1= (1,0,0),L2= (0,1,0),L3= (0,0,1).

Now, consider a lottery that gives youL1with probability¹₂, L2with probability ¹₄,

andL3with probability ¹₄. Definition

Compound Lottery:

Take anyLk ∈L,k =1, ...,K, lotteries defined overΩ.

Let(α₁, ..., α_K)be such thatα_k ≥0 andP

kα_k =1.

Then, a lottery that yieldsL_k with probabilityα_k is a compound lottery denoted by(L₁, ...,L_K;α₁, ..., α_K).

Lottery Types II

Consider a compound lottery denoted by(L1,L2,L3;¹₂,¹₄,¹₄), where L1= (1,0,0),L2= (0,1,0),L3= (0,0,1).

Now, consider

1 2L1+1

4L2+1

4L3= (1 2,1

4,1 4) Definition

Reduced Lottery: Take any compound lottery denoted by

(L1, ...,LK;α₁, ..., α_K). The lotteryα₁L1+...+α_KLK is called the reduced form lottery for these lotteries.

Illustrations

LetL={(p₁,p2,p3)|p_i ≥0andP pi =1}

Example

Simple Lotteries;L1= (1,0,0),L2= (0,1,0),L3= (0,0,1),L4= (¹₂,¹₂,0), L5= (¹₂,¹₆,¹₃)∈L.

Example

Compound Lotteries:(L₁,L₂,L₃;¹₂,¹₄,¹₄)and(L₄,L₅;¹₄,³₄).

both of compound lotteries produce Example

Reduced Lotteries:

1 2L1+1

4L2+1

4L3= (1 2,1

4,1 4) = 1

4L4+3 4L5

Preferences over lotteries I

If the decision maker has complete, transitive and continuous preference relation overL, then

∃U(.) :L7→Rsuch that

LL⁰⇔U(L)≥U(L⁰), LL⁰⇔U(L)>U(L⁰).

Definition

Expected Utility Form: U(.)has expected utility form if there existus∈R, s=1, ...,Ssuch that for everyL= (p1, ...,pS)∈L

U(L) =u1p1+...+uSpS. For example,U(1,0, ...,0) =u .1+0+...+0=u .

Preferences over lotteries II

Definition

von Neumann-Morgenstern (v.N-M)expected utility function:U(.) :L7→Ris v.N-M expected utility function if it has an expected utility form.

Take anyLdefined overS, and anyU(.) :L7→R.

Proposition

U(.) :L7→R has an expected utility form iff for any K lotteries Lk ∈L, k =1, ...,K and anyα₁, ..., α_K ∈R such thatα_K ≥0andP

kα_k =1, U(X

k

α_kL_k) =X

k

α_kU(L_k).

Preferences over lotteries III

Definition

Independence Axiom:defined onLsatisfies IA if for anyL,L⁰,L⁰⁰∈Land anyα∈(0,1),

LL⁰ ⇔[αL+ (1−α)L⁰⁰αL⁰+ (1−α)L⁰⁰].

That is, whenLL⁰the rankingαL+ (1−α)L⁰⁰αL⁰+ (1−α)L⁰⁰is independent ofL⁰⁰.

The Expected Utility Theorem I

Theorem

SupposeonLis rational, continuous and satisfies the IA, thencan be represented by a utility function that has an expected utility form, i.e.,

∃U() :L7→R and∃u₁, ...,u_S∈R such that for any L= (p₁, ...,p_S), L⁰= (p⁰₁, ...,p⁰_S)∈L,

LL⁰⇔U(L)≥U(L⁰),i.e,

LL⁰ ⇔

S

X

1

u_sp_s≥

S

X

1

u_sp⁰_s.

von-Neumann and Morgenstern (1944, Chapter 3)

The Expected Utility Theorem II

Proposition

Suppose U(.) :L7→R is a v.N-M utility function that representsonL, then U(.) :˜ L7→R representsonLiff there existβ >0andγ∈R such that

U(.) =˜ βU(.) +γ.