When you OTT to decide The Opportunity-Threat Theory of Decision-Making under Risk Mohan Pandey 56th Edwards Bayesian Research Conference March 1-3, 2018 Fullerton, California Greetings! I am delighted to be here amongst some of the finest brains. When Prof. Birnbaum first suggested that I come here and present my work, my immediate reaction was to develop cold feet. When you ought to decide though, both opportunity and threat matter. It looked more opportunity than threat and I requested Prof. Cavagnaro to accommodate me and he was kind enough to do that. So, here I am. My basic training is in pharmaceutical sciences and after my management degree my career has been in the area of R&D management. If you see some lack of rigor that’s because I have an out and out practitioners view of risk. If you see any value at all, it is because my early work landed in good hands of Prof. Jonathan Baron and through him Prof. Birnbaum. I cannot thank Prof. Birnbaum enough for his generosity and for bearing with me through 2017.

Am I risk-averse? Now, as far as I know, I am a human being. And, as far as the classical economic theory is concerned, I am supposed to be risk-averse? Essentially, everything else being constant, I’m supposed to prefer the safest option. That’s how I ought to behave. However, I don’t behave like that always. After all, I did decide to get married. On a more serious note, given what I do, I have routinely made investment decisions that are clearly not risk-averse. Investing in new drug discovery where odds of success are 1:10,000 is not risk-aversion. Well then, I’m told I’m being risk-seeking in that case. Which is fine, except for I start wondering if I’m schizophrenic. Is it one single thing to which I have two opposite responses, or is there some sense of coherence in how I think and may be I’m looking at two different things. This sense of discomfort is what drew me to this field of decision-making and behavioral economics. A practitioner was drafted into the realms of theories and ideas. March 2018 Mohan Pandey

Abstract The Opportunity-Threat Theory analyses risk into opportunity and threat components Thereby, it allows description of behavior as a combination of opportunity seeking and threat aversion. It can account for basic results as well as several “new paradoxes” that refuted cumulative prospect theory in favor of configural weight models. So, today I’d like to share with is a simple theory: Risk is not one thing. There is something called opportunity and something called threat. Risk is a situation where both opportunity and threat exist simultaneously. Without being schizophrenic, we are consistently opportunity-seeking and threat-averse. We take the plunge when opportunity seems to be bigger than threat. I’ll show that this simple theory is able to account for some basic results like the fourfold pattern as well as several new paradoxes that through Prof. Birnbaum’s body of work refute cumulative prospect theory in favor of configural weight models such as transfer of attention exchange. I hope it generates some good discussion and I carry home some new ideas. I am not an expert in the area. So, more importantly, I hope that this seed finds fertile ground elsewhere and is able to get a life. Looking for criticism, feedback, guidance, advice and collaboration! March 2018 Mohan Pandey

Structure Introduction Special OTT model (SOT) Results Discussion Simplified special opportunity-threat model (SSOT) Results Fourfold pattern Event-splitting effect Violation of stochastic dominance Violation of restricted branch independence Discussion Opportunity-Threat Theory Structural comparison of SSOT, CPT and TAX Future directions I’ll follow the format of my paper that got published in the latest issue of Judgment and Decision Making. After a brief introduction, I’ll describe a simple model of the theory and show how it explains fourfold pattern, event-splitting effect, violation of stochastic dominance and violation of restricted branch independence. The first one is of course that any theory in the field has to explain as a basic requirement. The other three lead to a variety of paradoxes. I will not go into details as such but only show how the new theory is able to account for them. Then, I’ll present the general theory. Briefly compare it with CPT and TAX and finally, with your permission, dabble into a bit of speculation. March 2018 Mohan Pandey

A brief and limited history of describing decision-making under risk Expected Value Theory (EVT) Expected Utility Theory (EUT) Allais Paradox Configural weight models including Transfer of Attention Exchange (TAX) Prospect theories including cumulative prospect theory (CPT) “New paradoxes” refute CPT in favor of TAX Pascal 1654, Huygens 1657 Bernoulli 1738, Neumann and Morgenstern, 1944 Allais, 1953 Birnbaum and others 1974, 1979, 1997 Kahneman and Tversky 1979, 1992 Birnbaum 2008 (summarized) To this audience, I don’t need to get into details of history. Suffice to say that, decision-making under risk is an important area and has received tremendous attention of scholars. It was becoming clear in 1950s that expected utility theory was not a good descriptive model and many different ideas attempting improving our understanding -- ranging from making minor to major modifications to the expected utility theory to proposing radical alternatives. Two branches stand out: first, the configural weight models and second, the prospect theory models. March 2018 Mohan Pandey

A simple insight People are influenced not only by the expected results of their action They are also affected by two components of risk Opportunity: the possibility of landing above reference Threat: the possibility of landing below reference 𝑉=𝑓(𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒, 𝑜𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑦, 𝑡ℎ𝑟𝑒𝑎𝑡) The theory that I propose is based on a rather simple insight. People are influenced not only by the expected results of their action. Final evaluation of a prospect depends on some reference appropriate in the context and then on an assessment of possibility of landing above that reference and an assessment of possibility of landing below the reference. March 2018 Mohan Pandey

Special Opportunity-Threat Model (SOT) For risky gamble, 𝑀= (𝑥 1 , 𝑝 1 ; 𝑥 2 , 𝑝 2 … 𝑥 𝑖 , 𝑝 𝑖 … 𝑥 𝑛 , 𝑝 𝑛 ), where, coefficients 𝛼 and 𝛽 represent psychological weights assigned to 𝜃 and 𝜓 respectively; and 𝜖 is an error term. 𝑉= 𝜇+ 𝛼.𝜃+ 𝛽.𝑏.𝜓+ 𝜖 𝜇= 𝑖=1 𝑛 𝑝 𝑖 𝑢 𝑖 One special model under this umbrella is SOT. 𝜃= 𝑢 −𝜇 (𝑛−1)/𝑛 𝜓= 𝜎 2 + 𝛼𝜃 2 1/2 𝑢 = 𝑖=1 𝑛 𝑢 𝑖 𝑛 𝑏= +1, 𝑖𝑓 𝜇<0, 𝑒𝑙𝑠𝑒−1 𝜎= 𝑖=1 𝑛 𝑝 𝑖 𝑢 𝑖 −𝜇 2 1/2 March 2018 Mohan Pandey

Simplified SOT 𝑉= 𝜇+ 𝛼.𝜃+ 𝛽.𝑏.𝜎 Assuming, 𝑢 𝑥 =𝑥 , non-extreme probabilities and no error 𝑉= 𝜇+ 𝛼.𝜃+ 𝛽.𝑏.𝜎 Limits 0<𝛼<1 −1<𝛽<1 𝛼= 1 8 𝛽= 1 3 “Priors” March 2018 Mohan Pandey

Fourfold pattern Expected value Observed value Risk aversion for gain of high probability Expected value Risk seeking for gain of low probability Risk aversion for loss of low probability Risk seeking for loss of high probability Observed value March 2018 Mohan Pandey

SOT explanation of fourfold pattern: Consider the case of binary gambles (x,p;0,1-p) 𝑉=𝑝𝑥 + 𝛼 1−2𝒑 𝑥 −𝛽 𝑝 1−𝑝 1 2 𝑥 𝑉=𝑝𝑥 + 𝛼 1−2𝒑 𝑥 + 𝛽 𝑝 1−𝑝 1 2 𝑥 March 2018 Mohan Pandey

SSOT, CPT and TAX, all three can explain the fourfold pattern Gamble Observed cash equivalent SSOT Components Calculated cash equivalent (V) 𝑥 𝑝 𝜇 𝛼𝜃 𝛽𝑏𝜎 Prior SSOT Prior CPT Prior TAX -$100 0.95 -84 -95 11 7 -77 -83 -59 0.05 -8 -5 -11 -9 $100 78 95 -7 77 59 14 5 9 10 8 Binary gambles from Tversky & Kahneman, 1992 March 2018 Mohan Pandey

Event-splitting effect 𝐺 𝑏𝑎𝑠𝑒 (𝑥 1 , 𝑝 1 ; 𝑥 2 , 𝑝 2 … 𝑥 𝑘 , 𝑝 𝑘 … 𝑥 𝑛 , 𝑝 𝑛 ), 𝑥 𝑖 >0 𝐺 𝑠𝑝𝑙𝑖𝑡 (𝑥 1 , 𝑝 1 ; 𝑥 2 , 𝑝 2 … 𝑥 𝑘 , 𝑝 𝑘 −𝑟; 𝑥 𝑘 ,𝑟… 𝑥 𝑛+1 , 𝑝 𝑛+1 ) There is no change in μ or 𝜎. However, 𝜃 𝑏𝑎𝑠𝑒 = 𝑛 𝑛−1 ( 𝑆 𝑛 −𝜇), 𝜃 𝑠𝑝𝑙𝑖𝑡 = 𝑛+1 𝑛 ( 𝑆 𝑛+1 + 𝑥 𝑘 𝑛+1 − 𝜇) where, S= 𝑥 𝑖 Thus, 𝜃 𝑠𝑝𝑙𝑖𝑡 −𝜃 𝑏𝑎𝑠𝑒 = 𝑥 𝑘 𝑛 − 𝑆− 𝜇 𝑛(𝑛−1) . Therefore, 𝑉 𝑠𝑝𝑙𝑖𝑡 > 𝑉 𝑏𝑎𝑠𝑒 if, 𝑥 𝑘 > 𝑆− 𝜇 (𝑛−1) . For a binary gamble (𝑥,𝑝;0,1−𝑝), 𝑥>0, splitting of higher branch satisfies this condition, 𝑥> 𝑥− 𝑝𝑥 (2−1) , and leads to increase in value. Splitting of lower branch, 0< 𝑥− 𝑝𝑥 (2−1) , leads to decrease in value, violating branch-splitting independence of Birnbaum (2007). March 2018 Mohan Pandey

Preference reversal predicted by SSOT and TAX, but not by CPT  Choice  % Choosing 2nd gamble Calculated cash equivalents Prior SSOT Prior TAX Prior CPT First gamble Second gamble First Second A: 85 to win $100 10 to win $50 05 to win $50 B: 85 to win $100 10 to win $100 05 to win $7 62 81.7 83.7 68.4 69.7 82.2 79.0 A’: 85 to win $100 15 to win $50 B’: 95 to win $100 26 78.1 75.7 62.0 Problems 1.1 (row 1) and 1.2 (row 2) from Birnbaum, 2008 March 2018 Mohan Pandey

Violation of stochastic dominance 𝑦 is a small positive quantity 𝑞 is a relatively small probability 𝐺 + (𝑥,𝑝;𝑦,𝑞;0,1−𝑝−𝑞) 𝐺 0 (𝑥,𝑝;0,1−𝑝) where, 𝑥>0 𝑧 is a positive quantity slightly lower in value to 𝑥 𝑠 is a relatively small probability 𝐺 − (𝑥,𝑝−𝑠;𝑧,𝑠;0,1−𝑝) There is negligible change in μ or 𝜎, however, 𝜃 + − 𝜃= 1−3𝑝 2 𝑥+ 1−3𝑞 2 𝑦− 1−2𝑝 𝑥 = −1+𝑝 2 𝑥+ 1−3𝑞 2 𝑦 ≈− 1−𝑝 𝑥 2 𝜃−𝜃 − = 1−2𝑝 𝑥− 1−3𝑟 2 𝑥+ 1−3𝑠 2 𝑧 = −𝑝 2 𝑥+ 1−3𝑠 2 𝑥−𝑧 ≈− 𝑝𝑥 2 These negative signs imply violation of stochastic dominance March 2018 Mohan Pandey

Calculated cash equivalents Violation of stochastic dominance predicted by SSOT and TAX, but not by CPT  Choice  % Choosing 2nd gamble Calculated cash equivalents Prior SSOT Prior TAX Prior CPT First gamble Second gamble First Second G + : 90 to win $96 05 to win $14 05 to win $12 𝐺 − :85 to win $96 05 to win $90 10 to win $12 73 70.6 74.9 45.8 63.1 70.3 69.7 G 0 : 90 to win $96 - 70.8 58.1 70.1 Problem 3.1 from Birnbaum, 2008 March 2018 Mohan Pandey

Violation of restricted branch independence 𝑆= (𝑥 1 , 𝑝 1 ; 𝑥 2 , 𝑝 2 … 𝑥 𝑖 , 𝑝 𝑖 … 𝑥 𝑛 , 𝑝 𝑛 ) 𝑥 𝑛 = 𝑦 𝑛 =𝑧 𝑝 𝑛 = 𝑝 𝑛 =𝑟 𝑅= (𝑦 1 , 𝑝 1 ; 𝑦 2 , 𝑝 2 … 𝑦 𝑖 , 𝑝 𝑖 … 𝑦 𝑛 , 𝑝 𝑛 ) Restricted branch independence assumes that a change in 𝑧 will not change the preference relationship between 𝑆 and 𝑅. Suppose 𝑆 is preferred over 𝑅, then, under SSOT, 𝑉 𝑆 > 𝑉 𝑅 . Then, if 𝜕 𝑉 𝑠 𝜕𝑧 ≥ 𝜕 𝑉 𝑅 𝜕𝑧 , 𝑉 𝑆 > 𝑉 𝑅 for all 𝑧. Otherwise, with increase in 𝑧, the gap in values will close and preference may get switched. March 2018 Mohan Pandey

𝜕𝑉 𝜕𝑧 = 𝜕𝜇 𝜕𝑧 +𝛼 𝜕𝜃 𝜕𝑧 −𝛽 𝜕𝜎 𝜕𝑧 =𝑟+𝛼 𝑛 𝑛−1 1 𝑛 −𝑟 −𝛽 𝑟 𝑧−𝜇 𝜎 𝜕𝑉 𝜕𝑧 = 𝜕𝜇 𝜕𝑧 +𝛼 𝜕𝜃 𝜕𝑧 −𝛽 𝜕𝜎 𝜕𝑧 =𝑟+𝛼 𝑛 𝑛−1 1 𝑛 −𝑟 −𝛽 𝑟 𝑧−𝜇 𝜎 Thus, if, 𝜕 𝑉 𝑠 𝜕𝑧 ≥ 𝜕 𝑉 𝑅 𝜕𝑧 , Then, −𝛽 𝑟 𝑧− 𝜇 𝑆 𝜎 𝑆 ≥−𝛽 𝑟 𝑧− 𝜇 𝑅 𝜎 𝑅 That is, 𝑧− 𝜇 𝑆 𝜎 𝑆 ≤ 𝑧− 𝜇 𝑅 𝜎 𝑅 Or, 𝑧≤ 𝜇 𝑆 𝜎 𝑅 − 𝜇 𝑅 𝜎 𝑆 𝜎 𝑅 − 𝜎 𝑆 Assume that for 𝑧′ this condition is not satisfied. Then, the corresponding second gamble (𝑅′) will be preferred over the corresponding first gamble (𝑆′), that is, a violation of type 𝑆𝑅′ will be observed. Note that this condition (𝑧≤ 𝜇 𝑆 𝜎 𝑅 − 𝜇 𝑅 𝜎 𝑆 𝜎 𝑅 − 𝜎 𝑆 ) is not dependent on 𝜃, 𝛼 or 𝛽 . March 2018 Mohan Pandey

Calculated cash equivalents Violation of restricted branch independence predicted by SSOT and TAX, but not by CPT  Choice  % Choosing 2nd gamble Calculated cash equivalents Prior SSOT Prior TAX Prior CPT First gamble Second gamble First Second S: 25 to win $44 25 to win $40 50 to win $5 𝑅: 25 to win $98 25 to win $10 40 17.9 17.2 20.0 19.2 19.7 28.1 S′: 50 to win $111 25 to win $44 𝑅′: 50 to win $111 25 to win $98 62 64.1 67.7 57.2 60.7 69.5 64.3 Problems 13.1 (row 1) and 13.2 (row 2) from Birnbaum, 2008 March 2018 Mohan Pandey

The Opportunity-Threat Theory: Illustration Future state S Objective state X Mental state M (1,1/6; 2,1/6; 3,1/6; 4,1/6; 5,1/6; 6, 1/6) Reference 3.5 Upside/downside 𝑀 𝑑𝑛 (1,1/6; 2,1/6; 3,1/6) (4,1/6; 5,1/6; 6, 1/6) 𝑀 𝑢𝑝 Relative Upside/downside 𝑌 𝑑𝑛 (-2.5,1/6; -1.5,1/6; -0.5, 1/6) (0.5,1/6; 1.5,1/6; 2.5, 1/6) 𝑌 𝑢𝑝 Assume agg(.) = EV(Ydn) + EV(Yup) and dis(.) = EV(Ydn) − EV(Yup) Then, Yagg = 0 and Ydis = −1.5. Thus, opportunity = 0 and threat = −1.5. Assuming V(.) = reference + opportunity + threat, V = 3.5 + 0 − 1.5 = 2. Thus, this gamble will be valued at $2.00 instead of expected value of $3.50. Aggregation/distance Opportunity/threat Valuation March 2018 Mohan Pandey

Structural comparison Consider, 𝑋= (𝑥 1 , 𝑝 1 ; 𝑥 2 , 𝑝 2 … 𝑥 𝑛 , 𝑝 𝑛 ), with 𝑛 exhaustive and mutually exclusive future states and 𝑝 𝑖 =1. Assume 𝑋 𝑢𝑝 = (𝑥 1 , 𝑝 1 ; 𝑥 2 , 𝑝 2 … 𝑥 𝑖 , 𝑝 𝑖 … 𝑥 𝑘 , 𝑝 𝑘 ) and 𝑋 𝑑𝑛 = (𝑥 𝑘+1 , 𝑝 𝑘+1 ; 𝑥 𝑘+2 , 𝑝 𝑘+2 … 𝑥 𝑗 , 𝑝 𝑗 … 𝑥 𝑛 , 𝑝 𝑛 ), such that the upside set has 𝑘 elements and the downside set has 𝑛−𝑘 elements. Define, 𝑉( 𝑋 𝑢𝑝 )= (𝑥 𝑖 − 𝜇), 𝑖=1 𝑡𝑜 𝑘 and 𝑉( 𝑋 𝑑𝑛 )= (𝑥 𝑗 − 𝜇), 𝑗=𝑘+1 𝑡𝑜 𝑛. Define 𝑉 𝑂 =𝑉 𝑋 𝑢𝑝 +𝑉( 𝑋 𝑑𝑛 ) analogous to 𝜃 and 𝑉 𝑇 = 𝑉 𝑋 𝑢𝑝 −𝑉( 𝑋 𝑑𝑛 ) analogous to 𝜎 (range instead of standard deviation). March 2018 Mohan Pandey

Two ranks, or, two Branches! Then, 𝑉= 𝜇+ 𝛼′.𝑉(𝑂)+ 𝛽′.𝑏.𝑉(𝑇) where, 𝜇= 𝑖=1 𝑛 𝑝 𝑖 𝑥 𝑖 and 𝛼 ′ and 𝛽 ′ are coefficients for respective terms. Now, assuming non-negative domain (𝑏=−1) and for simplicity 𝑉 𝑋 𝑂 −𝑉 𝑋 𝑇 >0 , expansion of all the terms yields, where, 𝑤 𝑖 = 𝛼′−𝛽′ + 1− 𝛼 ′ − 𝛽 ′ 𝑘−( 𝛼 ′ +𝛽′)(𝑛−𝑘) 𝑝 𝑖 and 𝑤 𝑗 = 𝛼 ′ +𝛽′ + 1− 𝛼 ′ − 𝛽 ′ 𝑘−( 𝛼 ′ +𝛽′)(𝑛−𝑘) 𝑝 𝑗 . 𝑉= 𝑖=1 𝑘 𝑤 𝑖 𝑥 𝑖 + 𝑗=𝑘+1 𝑛 𝑤 𝑗 𝑥 𝑗 Two ranks, or, two Branches! March 2018 Mohan Pandey

Future directions Nature and stability of coefficients Stochastic model Continuous probability distributions Nature and impact of error Investor behavior Other applications March 2018 Mohan Pandey

Key takeaways The Opportunity-Threat Theory analyses risk into opportunity and threat components Thereby, it allows description of behavior as a combination of opportunity seeking and threat aversion. It can account for basic results as well as several “new paradoxes” that refuted cumulative prospect theory in favor of configural weight models. March 2018 Mohan Pandey

Thank you Reference Pandey, M. (2018). The opportunity-threat theory of decision-making under risk, Judgment and Decision Making, 13(1), 33-41. http://journal.sjdm.org/17/17120/jdm17120.pdf Michael Birnbaum Jonathon Baron Konstantinos Katsikopoulis Bruce Car Daniel Cavagnaro Bristol-Myers Squibb Cover image: Author’s work mixed with https://commons.wikimedia.org/wiki/File:Scale_of_justice.png Early homo sapiens reconstruction image: Matteo De Stefano/MUSE https://commons.wikimedia.org/wiki/File:Homo_sapiens_-_Paleolithic_-_reconstruction_-_MUSE.jpg Dice images: Author March 2018 Mohan Pandey

Was he risk-averse? March 2018 Mohan Pandey

Backup material March 2018 Mohan Pandey

Derivation of 𝜃 and 𝜓 parameters March 2018 Mohan Pandey

Constraints Consider the case of binary gambles (𝑥,𝑝;0,1−𝑝), with 𝑥>0; here, 𝑛=2 and 𝜇=𝑝 𝑥. 𝜃= 𝑥 2 −𝑝𝑥 2−1 2 = 1−2𝑝 𝑥 𝜎 2 =𝑝 𝑥−𝑥𝑝 2 + 1−𝑝 0−𝑥𝑝 2 = 𝑝 1−𝑝 𝑥 2 Since 𝜇 is non-negative, 𝑏= −1 Therefore, 𝑉=𝑝𝑥+ 𝛼 1−2𝑝 𝑥−𝛽 𝑝 1−𝑝 1/2 𝑥 transforming to: 𝑉 𝑥 =𝑝+ 𝛼 1−2𝑝 −𝛽 𝑝 1−𝑝 1/2 March 2018 Mohan Pandey

Now, constraints 0< 𝑉 𝑥 <1 and 0<𝑝<1 are applied. They set the boundary conditions such that even for the smallest probability of smallest positive value of 𝑥, 𝑉 does not reduce to zero. Further, even for the smallest probability of 𝑥 not obtaining, magnitude of 𝑉 remains under certain 𝑥. Then, at 𝑝= 1 2 , 𝑉 𝑥 = 1 2 − 𝛽 2 . Thus, 0< 1 2 − 𝛽 2 <1 , which implies −1<𝛽<1. Also, for 𝑝~0, 𝑉 𝑥 ≈ 𝛼, thus 0<𝛼<1. March 2018 Mohan Pandey

Coefficients For estimation of 𝛽, consider a mixed outcome experiment with only two possible outcomes with equal probabilities ( 𝑥 1 , 1 2 ;− 𝑥 2 , 1 2 ) and observed 𝑉=0. Given symmetric distribution, 𝜃=0, giving, 𝑉=0= 𝜇− 𝛽.𝜎, or, 𝛽= 𝜇/𝜎. Now, 𝜇= 1 2 ∗ 𝑥 1 − 1 2 ∗ 𝑥 2 , and 𝜎 2 = 1 2 ∗ 𝑥 1 −𝜇 2 + 1 2 ∗ − 𝑥 2 −𝜇 2 = 1 2 ∗ 𝑥 1 − 1 2 ∗ 𝑥 1 + 1 2 ∗ 𝑥 2 2 + 1 2 ∗ − 𝑥 2 − 1 2 ∗ 𝑥 1 + 1 2 ∗ 𝑥 2 2 That yields, 𝜎= 1 2 ∗ 𝑥 1 + 1 2 ∗ 𝑥 2 . Thus, 𝛽= 𝜇/𝜎= 𝑥 1 − 𝑥 2 𝑥 1 + 𝑥 2 . Tversky & Kahneman (1992) reported four problems of this kind that yield 𝛽=0.42, 0.34, 0.34, 0.30. An average is taken and converted to an equivalent fraction for convenience, giving 𝛽= 1 3 . March 2018 Mohan Pandey

Now, with an estimate of 𝛽 at hand, 𝛼 can be estimated as follows. Assume that there exists a point where, 𝑉 𝑥 =𝑝. Then, 0= 𝛼 1−2𝑝 −𝛽 𝑝 1−𝑝 1/2 yielding, 𝛼= 𝛽 𝑝 1−𝑝 1/2 1−2𝑝 . Tversky & Kahneman (1992) considered 𝑝≤0.1 as low. Following that, 𝑝=0.1 is taken as the point of transition from low probability to moderate probability. Data from the same study shows that 𝑉 𝑥 ~𝑝 at =0.1 . At that point, with = 1 3 , 𝛼= 1 8 is obtained, which is in the range established above. March 2018 Mohan Pandey

Fourfold pattern: Calculation example Consider (100,5%;0,95%). The reference 𝜇=100∗5%=5. 𝜃= 1−2∗5% ∗100=90. With coefficient 𝛼= 1 8 the impact 𝛼𝜃= 1 8 ∗90≈11. This is positive and is taken as opportunity. 𝜎= 5%∗95% 1 2 ∗100=22. Since the gamble is in gains domain, 𝑏=−1. Thus, 𝑏𝜎=−22. With coefficient 𝛽= 1 3 we have 𝛽𝑏𝜎= 1 3 ∗ −22 ≈−7. This is negative and is taken as threat. The final value, 𝑉=𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒+𝑜𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑦+𝑡ℎ𝑟𝑒𝑎𝑡=5+11−7=9>5(𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒). As 𝑝 increases, 𝜃 decreases, crossing 0 when 𝑝= 1 2 , and turning negative after that. In the case of (100,95%;0,5%), 𝜇=95;, 𝜃= 1−2∗95% ∗100=−90, leading to negative impact of 𝛼𝜃= 1 8 ∗−90≈−11. Standard deviation and 𝑏 do not change so 𝑉=95−11−7=77<95(𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒). March 2018 Mohan Pandey

Moments-only models ruled out There is no change in μ or 𝜎 None of traditional measures of higher moments will change either. Since, for a moment of order 𝑚, 𝑝 (𝑥−𝑝𝑥) 𝑚 + 1−𝑝 0−𝑝𝑥 𝑚 = 𝑝−𝑟 𝑥−𝑝𝑥 𝑚 +𝑟 (𝑥−𝑝𝑥) 𝑚 + 1−𝑝 0−𝑝𝑥 𝑚 Thus, no moments-only model (for example, mean-variance or mean-variance- skewness models) will be able to explain change in value due to event-splitting March 2018 Mohan Pandey

Curve: CE/x vs. p CE values from Gonzalez & Wu (1999) March 2018 Mohan Pandey CE values from Gonzalez & Wu (1999)