Survey on Prospect Theory Kusdhianto Setiawan Gadjah Mada University

Expected Utility Theory •A prospect (x 1,p 1 ;…;x n,p n ) is a contract that yields outcome x i with probability p i, where p 1 + p 2 + … + p n = 1 •For simplification, lets omit null outcomes and use (x,p) to denote the prospect (x,p ; 0,1-p) that yields x with probability p and 0 with probability 1-p. >>>> u(0)=0 •The riskless prospect that yield x with certainty is denoted by (x)

Expected Utility Theory •Expectation: U(x 1,p 1 ;…;x n,p n ) = p 1 u(x) + … + p n u(x n ) that is, the overall utility of a prospect, denoted by U, is the expected utility of its outcomes. •Asset Integration: (x 1,p 1 ;…;x n p n ) is acceptable at asset position w if U(w + x 1,p 1 ;…;w + x n,p n ) > u(w) •Risk Aversion: u is concave (u” < 0)

Kasus 1 (Certainty Effect) Problem 1 Pilih antara: A.2500, P= (32%) 2400, P=0.66 0, P=0.01 B.2400, P=100% (pasti)67 (68%)* u(2400) > 0.33u(2500) u(2400) Problem 2 Pilih antara: C.2500, P= (77%)* 0, P=0.67 D.2400, P= (23%) 0, P= u(2400) > 0.33u(2500)

Kasus 2 (Invariance?) •Problem 3 Pilih antara: A. (4000, 0.8) atau B. (3000) 57 (58%)41 (42%) u(3000)/u(4000) > 4/5 (A, 0.8)(B, 1.0) •Problem 4 Pilih antara: C. (4000, 0.20) atau D. (3000, 0.25) 72 (73%)*26 (27%) (A, 0.25)(B, 0.25) •Mengurangi probabilitas dari 1.0 ke 0.25 memiliki efek yang lebih besar dari pengurangan probabilitas dari 0.8 ke 0.2

Kasus 3 (Certainty Effect, Invariance?) Problem 5 A.Kartu undian dengan peluang 50% menang, berhadiah wisata selama 3 minggu mengunjungi: Inggris, Perancis, dan Italia27 (28%) B.Tiket gratis liburan satu minggu di Inggris71 (72%)* Problem 6 C.Kartu undian dengan peluang 5% menang, berhadiah wisata selama 3 minggu mengunjungi: Inggris, Perancis, dan Italia65 (66%)* D.Kartu Undian dengan peluang 10% menang, berhadiah wisata selama satu minggu di Inggris 33 (34%)

Kasus 4 (Probable or Possible?) Problem 7 Pilih mana: A. (6000, 0.45) atau B. (3000, 0.9) 16 (16%)82 (84%)* Winning is more probable ! Problem 8 Pilih mana: C. (6000, 0.001) atau D. (3000, 0.002) 63 (64%)*35 (36%) Winning is possible but not probable! Substitution Violation! If (y,pq) is equivalent to (x,p), then (y,pqr) is preferred to (x,pr) where 0 < p,q,r <1.

Kasus 5 •Problem 3* •Pilih antara: •A. (-4000, 0.8) atau B. (-3000) 70 (71%)28 (29%) •Problem 4* •Pilih antara: •C: (-4000, 0.20) atau D. (-3000, 0.25) 32 (33%)66 (67%)*

Kasus 6 •Problem 7* •Pilih mana: •A. (-6000, 0.45) atau B. (-3000, 0.9) 85 (87%)*13 (13%) •Problem 8* •Pilih mana: •C. (-6000, 0.001) atau D. (-3000, 0.002) 41 (42%)57 (58%)

Reflection Effect P3: (4000, 0.8) < (3000)P3*: (-4000, 0.8) > (-3000) P4: (4000, 0.2) > (3000, 0.25)P4*: (-4000, 0.2) < (-3000, 0.25) P7: (3000, 0.9) > (6000, 0.45)P7*: (-3000, 0.9) < (-6000, 0.45) P8: (3000, 0.002) < (6000, 0.001)P8*: (-3000, 0.002) > (-6000, 0.001)

Implication of Reflection Effect •Risk aversion in the positive domain is accompanied by risk seeking in the negative domain •The preferences between positive prospects are inconsistent with expected utility theory, this is also the case for negative prospects. –Positive domain>>> the certainty effect contributes to a risk averse preference for a sure gain over a larger gain that is merely probable. –Negative domain>>> the same effect leads to risk-seeking preference for a loss that is merely probable over a smaller loss that is certain. •The reflection effect eliminates aversion for uncertainty and variability as an explanation of the certainty effect (see problem 3 & 4). –Certainty increases the aversiveness of losses as well as the desirability of gains.

Kasus 7 Problem 9 Andaikan anda ditawari untuk membeli polis asuransi kebakaran untuk rumah anda. Setelah menimbang premi yang harus anda bayar dan risiko yang harus anda tanggung jika terjadi kebakaran, anda tidak bisa menentukan apakah perlindungan yang diberikan asuransi sesuai dengan besarnya kerugian yang akan anda tanggung jika terjadi kabakaran (tidak punya preferensi, apakah akan beli asuransi atau membiarkan rumah anda tanpa perlindungan). Kemudian anda ditawari probabilistic insurance yang isinya sbb: Anda cukup membayar setengah dari regular premium insurance. Namun jika terjadi kebakaran pada tanggal ganjil anda harus membayar tambahan setengah dari regular premium dan perusahaan asuransi akan menanggung seluruh kerugian akibat kebakaran. Apabila kebakaran terjadi pada tanggal genap, anda akan mendapatkan kembali premi yang anda bayar dan kerugian anda tanggung sendiri. Akankah anda membeli probabilistic insurance tersebut? YA: 31 (32%)TIDAK: 67 (68%)*

Probabilistic Insurance •Reducing probability of a loss from p to p/2 is less valuable than reducing the probability of that loss from p/2 to 0. •Utility theory: if at asset position w one is just willing to pay a premium y to insure against a probability p of losing x, then one should be willing to pay a smaller premium ry to reduce the probability from p to (1-r), where 0<r<1. •Formally: if one is indifferent between (w-x,p ; w,1-p) and (w-y), then one should prefer probabilistic insurance (w-x, (1-r)p; w-y,rp;w-ry, 1-p) over regular insurance (w-y).

Kasus 8 Problem 10 Bayangkan anda mengikuti sebuah quiz di TV yang terdiri dari dua babak. Pada babak pertama, terdapat peluang 75% untuk mengakhiri babak pertama tanpa mendapatkan hadiah dan peluang 25% untuk maju ke babak kedua (terakhir). Jika anda berhasil maju ke babak kedua anda diberi pilihan sbb: A.Menjawab pertanyaan quiz dengan hadiah $4000 dengan peluang menjawab benar 80%.65 (66%)* B.Mundur dari babak kedua, dan menerima $ (34%) Tentukan jawaban anda sebelum melangkah ke babak kedua !

The Isolation Effect •In problem 10, one has a choice between 0.25 x 0.8 = 0.2 chance to win 4000, and a 0.2 x 1.0 = 0.25 chance to win •In terms of final outcomes and probabilities one faces a choice between (4000, 0.20) and (3000, 0.25). This is similar to problem 4!

Problem 41/4 3/4 1/5 4/ Problem 10 1/4 3/4 4/5 1/ Risky prospect Riskless prospect Risky Prospect Risky prospect

Kasus 9 Problem 11 Sebagai tambahan atas berapapun uang yang anda miliki sekarang, anda saya beri $1000. Kemudian anda ditawari dua buah game, mana yang akan anda pilih: A. (1000, 0.50) atau B. (500) 48 (49%)50 (51%) Problem 12 Sama spt di atas, tapi anda saya beri $2000. Kemudian anda diharuskan memilih satu dari dua permainan yang tidak anda kuasai, mana yang anda pilih: C. (-1000, 0.5) atau D. (-500) 50 (51%)48 (49%)

Problem •In terms of final outcomes, both problems can be stated as: A = (2000, 0.5 ; 1000, 0.5) = C B = (1500) = D •Implication: –Subject did not integrate the bonus with the prospect. –State of wealth (final asset position) is less matter than gains and losses (changes of wealth) ---- violation of utility theory! –Carriers of value or utility are changes of wealth, rather than final asset position that include current wealth.

PROSPECT THEORY •Two Phases in the choice process: –Early phase of editing –Subsequent phase of evaluation •Editing Phase: –Coding (determine gains/losses & reference point/current asset position) –Combination: (200,.25; 200,.25) = (200,.50) –Segregation: (300,.80; 200,.20) = (200; 200,.80) –Cancellation (for a set of two or more prospects) i.e: Choice between (200,.20; 100,.50;-50,.30) & (200,.20; 150,.50; -100,.30) can be reduced become (100,.50; -50,.30) & (150,.50;-100,.30)

VALUE •The decision maker is assumed to evaluate each of edited prospects, and to choose the prospect of highest value. •The overall value of edited prospect denoted V, is expressed in terms of two scale, π and ν. •π : the impact of p on the V, but π is not a probability measure (π(p) + π(1-p) < 1) •v : the subjective value of outcome x = v(x)

Axioms •If (x, p; y, q) is a regular prospect (i.e. either p + q < 1, or x ≥ 0 ≥ y, or x ≤ 0 ≤ y) then V(x, p; y, q) = π(p)v(x) + π(q)v(y) •v(0)=0, π(0)=0, π(1)=1 •V(x, 1.0) = V(x) = v(x)

Axioms •Strictly positive and strictly negative prospects follow a different rule •If p + q = 1 and either x>y>0 or x<y<0 then V(x, p;y, q) = v(y) + π(p)[v(x) – v(y)] •Meaning: the value of a strictly positive or strictly negative prospects equals the value of riskless component plus the value-different between outcomes, multiplied by the weight associated with the more extreme outcome.

Example of Strictly Positive Prospect V(400,.25;100,.75) = v(100) + π(.25)[v(400) – v(100)] The essential feature of above equation is that a decision weight is applied to the value-difference v(x) - v(y), which represents the risky component of the prospect, but not to the v(y), which represent the riskless component.

The Value Function VALUE LOSSESGAINS Referrence point

The Weighting Function (π) Stated Probability: p Decision Weight: π(p)