The Prospect Theory and First Price Auctions: an Explanation of Overbidding

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Introduction
The prospect theory was introduced as a critique of the expected utility theory as a decision-making model under risk, in a paper published in Econometrica in 1979 by Kahneman and Tversky entitled "Prospect Theory: An Analysis of Decision under Risk".This paper made a significant contribution since it showed that people are systematically violating the properties of the expected utility model, which at the time was the workhorse model for decision-making under uncertainty.However, the prospect theory has not been applied in economic theory, not because it was irrelevant outside the laboratory setting, but because it was hard to know how to apply it, see Barberis (2013).Mostly, researchers from the behavioural economics, i.e. behavioural finance, are involved with the application of the prospect theory.Studies in finance are conducted through the CAPM models of Sharpe (1964) and Lintner (1965)  1 , where investors are evaluating utility in accordance with the expected utility, stating that securities with "higher betas", i.e. the returns of the securities that covary more with the return of the overall market.Although research by Fama, and French (2004) concluded that this model does not receive empirical support 2 , the Black (1972) model 3 assumes no riskless asset.This version of the CAPM model 4 is more robust in empirical testing.First, the potential shortcoming of the original CAPM model is that the 'true' market portfolio is unobservable (see : Roll, 1977).Roll (1977) also pointed to mean variance tautology, namely that mean-variance efficiency and the capital asset pricing model equation are mathematically equivalent 5 .Hence, this raised the question of whether one can do better in explaining cross-section average returns using a model in which investors evaluate risk in a psychologically plausible way.Barberis and Huang (2008) studied the asset prices in a one-period economy in which investors derive prospect utility from the change in the values of portfolios.The study found that a security whose return distribution is right tail skewed will be overpriced, relative to an economy with expected utility investors, and will earn a lower average return.Previously, papers used pricing of financial securities when investors make decisions according to the cumulative prospect theory (CPT) of Tversky and Kahneman (1992).Under the cumulative prospect theory, one uses a value function defined over gains and losses, concave over gains and convex over losses, and kinked in the origin, and using weighted probabilities.Furthermore, this study explains that overweighing the tails is a modelling device for capturing the common preference for a lottery-like, positively skewed wealth distribution.Kőszegi and Rabin (2006, 2007, 2009), in their respective papers proposed how to apply the prospect theory in economics, suggesting that the reference point people use to compute gains and losses are their rational expectations and beliefs6 .In the case of finance, several authors and publications using different techniques to measure the skewness concluded that more positively skewed stocks will have lower on average returns, see : Boyer, Mitton and Vorkink (2010), Bali, Cakici, and Whitelaw (2011).The prospect theory implies that stocks involved in an offering should have lower average returns.Green and Hwang (2012), found that the higher the predicted skewness of an initial public offering stock, the lower its longterm average return.An open IPO, which is a modified Dutch auction (strategically equivalent to a First Price Auction), based on an auction system designed by Wiliam Vickrey 7 .This auction method ranks bids from highest to lowest, then accepts the highest bids that allow all shares to be sold, with all winning bidders paying the same price, similarly to T-bills auction.One consistent outcome found in the experiments involving independent private value first price auctions8 is that the subjects consistently bid above the risk neutral Nash equilibrium (RNNE) bid, see: Dorsey and Razzolini (2003).For a well-known example of "misbehavior" of bidders in FPA auctions, see: Harrison (1989); Cox, Smith and Walker (1988).Next, the authors examined to what extent the prospect theory is able to explain overbidding in first price auctions (FPA).Research concluded that overbidding occurs when bidders are riskseeking when faced with a risky choice leading to losses, or risk-averse when faced with a risky choice leading to gains cf.Kirchkamp and Reiss (2004); Kagel andLevin (2002, 2016).Earlier studies explaining the overbidding with CRRA constant relative risk aversion include (Cox et al., 1982(Cox et al., , 1983a(Cox et al., , 1983b(Cox et al., , 1984(Cox et al., , 1985)).Yet, the earlier literature states that risk aversion cannot be the only factor behind overbidding (cf.Kagel and Roth, 1992).Loss aversion was the main evidence for overbidding in a multidimensional reference dependent model for FPA and SPA (see : Lange, & Ratan 2010).Some studies used the prospect theory in their explanation of overbidding in auctions, examples include (Goeree et al., 2002), that used subject probability weighting function (PWF) suggested by Prelec (1998).Ratan (2009) also used the same PWF, together with a multidimensional reference-dependent model.Armantier and Treich (2009a;2009b) stated that any star-shaped probability weighting function is able to explain overbidding.I In addition, this paper proves that when overbidding occurs in first price auctions, the results are for symmetric and asymmetric auctions, and also that certainty equivalent function is convex in this case, which implies a risk--seeking utility function.

The prospect theory value of a game
Let us consider a game with two possible outcomes:  with probability  and  with probability 1 − , where  ≥ 0 ≥ .The prospect theory value of the game is equation 1 In prospect theory the probability of weighting  is concave and first order convex, e.g.equation 2 For some ∃ ∈ (0,1).A useful parametrisation of the prospect theory value function is power law function equation 3 A fourfold pattern of risk aversion  is: 1. Risk aversion in the domain of likely gains.2. Risk aversion in the domain of unlikely gains.3. Risk seeking in the domain of likely losses.4. Risk seeking in the domain of unlikely losses.Some properties of the prospect theory value functions are: • They are scale invariant, i.e. ∀ > 0 Now, let us consider two gambles (uncertain outcomes9 ), the second gamble being scaled by  equation 4 If someone prefers  to  ′ then they will prefer  to  ′ for  > 0, and if ,  ≥ 0, then: If  ′ ,  ′ ≥ 0, and someone prefers − to − ′ , the question that arises is of the robustness of the results, and the results are: very robust 1.where there is loss aversion at reference point,  > 1 medium robust 2. where there is convexity of  for  < 0 slightly robust 3. dependent on the underweighting or overweighting the probabilities () ≷ .
In application, often a simplified version of the PT theory is used: Now, let us consider gamble  and − with 50:50 chances.The question arises here as to what risk premium Π would agents pay to avoid small risk .It can be proved that as  → 0 this premium becomes ( 2 ), and this is called a second order risk aversion.In fact, it can be shown that for twice continuously differentiable utilities equation 11 where  is the curvature of  at 0 that is  = −  ′′  ′ .Now, let us take an agent with wealth , and this agent takes the gamble if equation 12 i.e.Π ≥ Π * , where equation 13 Now, let us assume that  is twice differentiable and take Taylor expansion of (Π) for small  and Π: equation 14 Now, using the definition of (Π * ) = () to get: and to solve: . First, find the roots of equation 18 thus the roots are equation 20 As when there is no risk, the risk premium should be 0, then the relevant root is Then, take the Taylor expansion for small  equation 21 Now, let us remember that First order risk aversion of prospect theory Let us consider a gamble as for expected utility.One takes the gamble if Π ≥ Π * , where � (Π * + ) + (0.5)(Π* − ) = 0 to show that in the prospect theory, as  → 0, risk premium Π is of order  when reference wealth  = 0; this is called first order aversion.Now let us compute Π for () =   and  = −||  .Premium Π at  = 0 satisfies: � and use the fact that − + Π * < 0 to obtain equation 24 then equation 25 where  is equal from the second order risk aversion to equation 26 Then equation 27 but empirically  value will be equation 28 Note that when  = 1 , the agent is risk neutral and the risk premium is 0. Next, two extensions of the prospect theory are presented.First, both outcomes are positive 0 <  <  , then equation 29 For the negative gambles, apply the same formula and 0 >  > .Continuous gambles' distribution for expected utility gives equation 30 The prospect theory gives Kahneman, Knetsch, and Thaler (1990) showed that in expected utility  =  , or willingness to pay is equal to willingness to accept 10 .Otherwise, as in Sugden (1999), and in Horowitz and McConnell (2003): is labelled as the income effect.Horowitz and McConnell (2002) found that WTA is about seven times higher than WTP.Hanemann (1991), showed that the difference between WTP and WTA depends on the ratio of ordinary income elasticity of demand for the good with respect to the Allen-Uzawa elasticity of substitution between the 10 Willingness to accept is the minimum amount of monetary units that а person is willing to accept to abandon a good or to put up with something negative, such as pollution.
good and a composite commodity (see: (Hicks & Allen, 1934 a,b;Uzawa, 1962)).The elasticity of substitution can be defined as equation 32
However, if  0 → 0 then if there are very few substitute products of  in  this generates values  → ∞.Another association here is equation 46 Contributions in the literature include Knez, Smith and Williams (1985), who argued that the difference between buying and selling prices can be attributed by the thoughtless application of normally sensible bargaining habits, namely understanding one's WTP and overstating the minimum acceptable price at which one would like to sell (WTA).Coursey, Hovis and Schultze (1987), stated that the discrepancy between WTP and WTA diminished with the experience of the market setting14 .

Risk seeking
Let us take stock market return function as  =  +  ,where  ∼ (0,1) is a return of the 'gambles' or actions with uncertain outcomes.Investors invest in proportion  in stock and with proportion 1 −  in a riskless bond with return 0. Total return is then Since  =   for positive and  = −||  , using the homotheticity15 one obtains: Thus, from previous optimal  = [0, ∞] ,depending on the sign of the last integral.This is a problem because one does not have a concave function, and without concavity it is easy to have those instant solutions.One solution here is that   +   or the expected utility value function added up with the prospect theory value function.Now if one implicitly takes reference point   to be the wealth at  = 0, then the gamble is  0 + : equation 51 Barberis, Huang and Santos (2001) also drew on the prospect theory and proposed a new asset pricing framework that is derived in part by the traditional consumptionbased approach (see Lucas, 1978), but also incorporated the prospect theory of Kahneman and Tversky (1979), and insights by Thaler and Johnson (1991).Now, welfare is hard because it depends on the time frame.Let us take the integrated and separated prospect value functions: equation 52 The costs of business cycles are given as   =  +   ,where  is average monthly consumption, with normal iid   with   = 0, if one takes   =  = 0.With the prospect value integrated over one year and the segregated one given as equation 53 In the expected utility, welfare is defined as  = ( +   ), measure of welfare loss due to business cycle is imbedded in   by fraction  of the consumption that people would accept to give up in order to avoid consumption variability, and  solves: then consumers in accordance with the prospect theory value more the stability of consumption around the reference point where they are first order risk averse, and their risk aversion depends on their horizon.

Cumulative prospect theory
In 1992, Tversky and Kahneman proposed a new theory known as the cumulative prospect theory.The prospect similar to the prospect theory is denoted by (, ) where -are the outcomes of the prospect and  are their respective probabilities.The reference point is defined as  0 = 0, and all other outcomes are defined in terms of the reference point.A prospect with  +  + 1 outcomes is given by ( − ,  − ; … … ; .  ;   ), where ,  ≥ 0 , and  − ≤ ⋯ . .≤   .Now, to denote a prospect with  and  + , which is the positive part of the prospect ( 1 ,  1 , … ,   ,   ), and  − is the non-positive part of the prospect ( − ,  − , … .,  0 ,  0 ), then the value of the prospect is given as which is separated in terms of gains and losses.If all of the outcomes in the prospect are positive, then () =  + ( + ), and if all the outcomes are negative, then () =  − ( − ).The values of the positive and negative outcomes are given as: It is assumed that there exists a strictly increasing value function that satisfies :  → ℝ, satisfying ( 0 ) = (0) = 0 and  + ( + ) = ( 1 + , … .,   + ), and  − ( − ) = ( − − , … .,0), the decision weights or the probabilistic distortions for gains are given as: equation 57 Similarly, the decision weights for losses are given as: where  + (0) =  − (0) = 0 and  + (1) =  − (1) = 1, because if something is impossible it should not impact on individual preferences, and when something is certain to happen, then the effect should be the value that the outcome is given.Hence,   are decision weights or probability distortion functions, and  + ,  − are the decision weighting functions.Therefore, now the value of a prospect is given as: Prospect (, )is more preferred than (, ) so that (, ) > (, ) and is indifferent when (, ) = (, ).Thus, for decisions under risk one has: weighting functions for gains  + and losses  − can be defined as mapping that assign event   to a space denoted as  ,as a number between 0 and 1.The previous satisfy  − (∅) =  + (∅) = 0 and  + () =  − () = 1 , and  + (  ) ≤  + �  �;  − (  ) ≤  − �  �; ∃  :   ⊃   where 1.
The decision weighting functions  + and  − are defined by the previously mentioned properties above, but they are not directly observable (see: (Wakker & Tversky, 1993)).Therefore, a two-stage decision process is assumed, following: Tversky and Fox (1995); Fox et al. (1996); Kilka and Weber (2001).Fox and Tversky (1998) proposed specifying the weighing function by a two-stage approach: where  is a weighting function,  are probability judgments following the support theory (see Tversky and Koehler (1994)),   is the event considered, and   is the probability weighting function under risk.Wakker (2001) gave a formal justification for this decomposition of the weighting function.The axiomisation of the cumulative prospect theory (CPT) was presented in (Wakker & Tversky, 1993;Chateauneuf & Wakker, 1999).The value function exhibits the same properties as fin the prospect theory, i.e. reference dependence, diminishing sensitivity, and loss aversion.Hence, () is concave above the reference point ′′ ≤ 0;  ≥ 0, and convex above the reference point  ′′ ≥ 0;  ≤ 0. The previous reflects diminishing sensitivity, which means the impact of changes in the domain of gains and losses diminishes when the distance from the reference point increases.The value function is steeper for losses than for gains, i.e.  ′ () <  ′ (−);  ≥ 0, since losses persist longer than gains, there is also extensive experimental evidence that losses have greater impact on preferences than gains (cf.Tversky & Kahneman, 1991).The parametric form of the value function proposed by Tversky and Kahneman (1992) is equation 61 The decision-weighting function takes cumulative probabilities and weights them nonlinearly.This means that a change in probability from 0.1 to 0 and from 0.9 to 1 has more influence than a change in the probability from 0.4 to 0.5.Small probabilities tend to be overweighted, and moderate to high probabilities tend to be underweighted.This give rise to an S-shaped function concave near 0, and convex near 1 (inverse S-shaped).Function () exhibits subadditivity 16 if ∃ 1 ; ∃ 2 , such that equation 62 The following two equations give the parametric form proposed by Tversky and Kahneman (1992) equation 63 The value function where loss aversion parameter  is given as equation 64 where () and () are defined as follows: Another parametric form17 which serves as a weighing function was proposed by Gonzalez and Wu (1999) equation 66 The curvature parameter is the previous function  which represents the degree of diminishing sensitivity, the degree of curvature increases as  ≫ 0 increases.
The elevation parameter represents the attractiveness of the bet and the elevation increases as  ≫ 0 or  → ∞.Another weighting function was proposed by Prelec (1998) equation 67 () =  (−)   > 0 and the point at which the weighting function crosses the line () =  is fixed at 1  ≈ 0.36788.The inflection point for the parametric equations typically occurs at  < 0.4.The function exhibits the inverse s-shape, concave for lower probabilities and convex for the upper probabilities.

Methods of estimation
The methods described here include local search optimisation and the nonlinear squares approach.The search space for both parameters (alpha and gamma) was restricted between 0 and 1.With a small search space, it is easier to find the parameters that give the smallest MSE (mean-squared error).With the local search optimisation one can constrain the values of the parameters.

Nonlinear least squares
This method follows the method introduced by Scales (1985), and Aster et al. (2005).It is known to minimise the weighted sum of the square of the residual.equation 68   () = (  , ; ) − (; ),  = 1 …    () is the residual, while  is the invested share (vector of parameters to be estimated), (; ) is the value of certainty equivalent (CE)18 for the prospect, and   = ℎ  (, , ), is the value of each outcome in the sample.Now, Ε() is the weighted sum of the residuals: where   is the weight associated with each sample .If () = √() where () = ( 1 (), … ,  2 ()  , and the weights are given with the following diagonal matrix matrix 1 The minimum of Ε() is given when the gradient is equal to zero: where () is a Jacobian matrix of (), and thus one can denote matrix 2 By Newton's iteration method, the search vector for the  ℎ iteration is given as   and   is the value for  in the  ℎ iteration equation 70 Expanding ∇Ε(θ) by using the Taylor series with respect to  about  +1 =   one obtains equation 71 where  2   () is the Hessian matrix of   () which is given by matrix 3 Newtons' iteration method (see e.g.(Amparo et al., 2007)), in general is given as Now if: the value function for any given lottery is given as where  0 is the reference point.The consequences above  0 are considered gains, and ones below are losses.In the previous expression  is a probability weighting function, and  is any given lottery.The cumulative density functions for losses are given as equation 75 and the corresponding cumulative density function for gains is given as The agent is evaluating the consequences, having in mind the reference-dependent utility function equation 77 The probability weighting function is given as () =  −(−)  , just as in the Allais paradox (see: Allais

Results from the MATLAB simulation
The results from the MATLAB simulation using the code by Pitcher (2008) are shown below (Figure 1).where:   The previous matrix is multiplied with the weights scalar so to obtain the value function: As for the weight functions (see (Currim & Sarin, 1989)), proposed by Prelec (1998), defined as :

First Price Auctions and high bidding (overbidding)
The bidding strategy in First Price Auctions (FPA) 19 is given as: 0 <  < 1 or zero otherwise.In the previous integral s is a signal, expression (  ) represents buyers i valuation of the object bid that wins the object, and in such case   represents bidders i reservation value.In auctions, each bidder calculates his/her winning probability by compounding the probabilities that every other bidder bids less than his/her bid.In equilibrium, any bidder  ∈  with valuation   has expected payoff: equation 85 when the CRRA coefficient is being set, i.e. when bidders are not risk neutral, the corresponding expected payoffs (and corresponding revenue) of the bidders are given as: equation 86 If  is CRRA coefficient then: equation 87 The generalised FPA-with reserve price bid is given as (cf.(Krishna, 2009)): equation 88 In the previous expression  signals are drawn from private values distribution v, so   =   .In the CRRA case utility is given as () =  1− , and now the bid function is In the previous expression  0 <  < 1 or zero otherwise.In the case where the reserve price is set  > 0: Now if the coefficient is CARA (constant absolute risk aversion), i.e. if the utility function is given with the following expression () = 1 −  − where  > 0, then the according bidding function is given as: where w represents wealth of the bidder, approximated by his/her valuation of the object which is subject to bidding at the auction.The inverse of equilibrium bidding strategy (Maskin & Riley 2000;Fibich & Gavish, 2011) is given as: Bidders submit bids that are solutions to the optimisation problem, as in (Gayle & Richard, 2008): The probabilities of winning are equation 93 In the previous expression ℓ  () =   (  ()), and  represents the reserve price in auction.Expected revenue for the auctioneer is equation 94 and the expected revenue for the bidders  group is equation 95 In the prospect theoretical approach, bidders weight probabilities, and they evaluate gains of the lottery relative to theory reference point via the value function.

Proof of proposition 1
Here, FOC with respect to  is equation 99 In symmetric equilibrium  = (  ), hence equation 100 which, when arranged yields: If one multiplies the whole expression with ( − 1)�(  )� −2 one obtains equation 102 one can see clearly that   <   ; ∀   ∈ (0,1).By differentiating   with respect to   one concludes that it is increasing in   since equation 104 > 0 Now, when bidders are bid-shading or pretending that   ∼   , and this bidder's expected payoff is () <   , then for competitors  equation 105 and from the assumption that F (CDF) is a uniform distribution.one has: where function  is defined as: : (0,1) → (0,1).This result applies ∀ ∈ (0,1), also in the previous expression  ∈ (0,1).Now, for the equilibrium analysis take any bidder  ∈ , assuming again that any bidder  ∈   ̅ {} bids according to symmetric, differentiable strategy  ∈   , the expected payoff of bidder  from bidding bid  ∈ (0, ) is given as: where  1 = 1 if  ≤   − (  ) and is equal to  if otherwise.Hence, under the previous conditions a new proposition follows.
Proposition 2 A unique risk-neutral symmetric equilibrium in FPA auctions is characterised by ; 0  when bidders are assumed to weight probabilities after compounding and they are evaluating payoffs relative to the reference point.

Proof of proposition 2
Here, FOC with respect to  is equation 110 The equilibrium bidding strategy is  = (  ) , and thus equation 111 By arranging the previous expression equation 112 for the optimal bid one has: In the previous expression () −1 = () and one can replace the reference point  with  =   20 .Now it can be checked whether equilibrium ∃ >   − (  ) and whether ∃ =   − (  )2 0F 21 .Then the previous equality becomes inequality equation 114 The previous concludes that there is no interior solution.Hence, (  ) =   − (  ) if one assumes that  ≥   − (  ).Now, assuming that  ≤   − (  ), to check whether  ≤   − (  ) when  = 1.Therefore equation 115 The previous expression for high values of  and   means that (  ) >   − (  ).If F is uniform distribution, then there exists equilibrium (cf.Reny (2011).Under certain conditions such as when the player set of strategies is non-empty and closed, the density function should be continuous, the type set should be partially ordered, the strategy set needs to be a compact metric space and a semi-lattice with a closed partial order, and the utility function should be measurable and bounded.This article generalised Athey's (2001) andMcAdams' (2003) results on the existence of monotone pure strategy equilibria.According to the previous strategy, bidders are allowed to bid as high as they want so that the utility function is not bounded, which means that the strategy sets are not compact either (see also (Keskin, 2011)) 22 . 20Corresponding expression would be If one takes that this holds:  = { ∈ ℬ|∃ ∈ :  ⊆ },where Β is a topological basis of  and Ο is an open cover of .Here  is a refinement of , and ∀  ∈ , and one select   ∈ , then  = {  ∈ | ∈ }, see (Munkres, 2000).
However, these two properties can be satisfied when one narrows the strategy set to [0,1].Hence, if  is a uniform distribution then proposition 2 holds.∎ The definitions and proof of stochastic dominance and overbidding are given in Appendix.

Error return function (erf) for the prospect theory
There exists a market with -number of securities.At some point in time  an investor can purchase   units of generic  ℎ security23 .The allocation is represented by the  ℎ dimensional vector , where   is a price at the generic time  of the generic  ℎ security.Now, given this and with allocation vector , the investor forms portfolio equation 116 In investment horizon , the market prices of the securities are multivariate random variables.The simple function of a one-dimensional random variable price is equation 117 The objective of the investor is  , as he/she wants the largest possible amount of benefits according to the non-satiation principle.Absolute wealth is given as equation 118 In relative wealth, the investor is concerned with overperforming a reference portfolio, whose allocation is denote with , and the objective of maximisation is equation 119 In the previous equation,  is a normalisation factor which means that at the time when the investment decision is made, the reference portfolio and the allocation have the same value equation 120 According to the prospect theory (Kahneman & Tversky, 1979), investors are concerned with changes in wealth rather than in absolute wealth.Thus, their objective becomes equation 121 The explicit expression in terms of allocation becomes:   =  ′ ( + −   ).With market vector , the introduced previous expression becomes   =  ′ , where  ≡  +  + .Furthermore,   =  ′  + where  ≡   −   ′  ′   where   is an identity matrix.Hence, it follows that the allocation function is homogenous of degree one   =   , and also is an additive function  + =   +   .When ranking two allocations , , the two allocations might not be comparable, therefore all the features of allocation are summarised as :  ↦ ().Now the investor can chose the allocation with the highest degree of satisfaction.This approach is different from the stochastic dominance approach (see: (Ingersoll, 1987;Levy, 1998;Yamai & Yoshiba, 2002)) 24 .For the features of satisfaction see also Frittelli, Rosazza and Gianin (2002).The scale invariant index is also known as Sharpe ratio defined as () = {  } .{  } , which means that high standard deviation is a drawback if the expected utility is positive.Additionally, monotonicity requirements are as follows:   ≥   ; ∀ ⇒ () ≥ ().For sensibility, i.e. monotonicity, see Artzner, Delbaen, Eber, and Heath (1999).A further requirement is that of positive homogeneity:   =   , ∀:  ≥ 0, and additive ( + ) ≥ () + (), or sub-additive ( + ) ≤ () + ().Next, regarding concavity and convexity -an index of satisfaction is said to be concave for ∀:  ∈ (0,1) and the following inequality holds: Similarly, the index of satisfaction is said to be convex for ∀:  ∈ (0,1) when the following inequality holds ( + (1 − )) ≤ () + (1 − ).On the opposite side of the satisfaction risk premium is dissatisfaction due to the uncertainty of risky allocation  ≡ () − ( + ) where  is risk-free allocation and  is any fair game.Risk aversion is if () ≥ 0, while risk propensity is () ≤ 0.

Certainty equivalent
Let us consider the expected utility from a given allocation 24 Strong dominance is    −  (0) ≡ ℙ�  −   ≤ 0� = 0; this is a strong dominance or order zero dominance, and weak dominance when    () ≤    (), ∀:  ∈ (−∞; +∞), this is also called first order dominance.where  has a dimension of money and it cancels it out.These properties are depicted below (Figure 2):

Translation invariant
The objective of the investor, besides being positive and homogenous, is additive25 , and by adding two portfolios , , one obtains a sum of the two separate objectives equation 125 The utility from the sum of the two alternatives is unrelated with the satisfaction drawn by the investor from investing in separate portfolios.Thus, the corresponding index of satisfaction is given as: The previous property is called translation invariance (cf.Meucci, 2005).One can restate the translation invariance property as equation 127 Proof: Let (  ) =1 ∞ be a sequence of open intervals (an open interval is an interval that does not include endpoints).The open interval {:  <  < } is denoted (, ) (Gemignani, 1990).Then (  +  ) =1 ∞ is a sequence of open intervals that cover  + , and therefore equation 128 , and now  * () ≤  * ( + ).From  * ( + ) ≤  * () and  * () ≤  * ( + ) one can conclude that  * () =  * ( + )∎.
The relative strength ratio is a possible way of comparing bidder's  beliefs about his/her rival with bidder's  beliefs about the rival is through the relative strength ratio   , see (Kirkegaard, 2009) ;  , () =   () () .
where in previous expression,    () and    () are the respective probabilities of winning the auction.The bidders are equally well-off at ̅ so that  , () = 1.Whereas in the previous expression The result holds if all the bidders subjectively weight probabilities with the same inverse S-shaped PWF. , and positive when  = 1, and thus any bidder with valuation  underbids if  <   * , and the bidder overbids if  >   * ∎ (see also (Keskin, 2015)).
(1953)), weighted utility theory is given as equation 78 () = � (|, )() :  → ℝ.The probabilities must be distorted if one wants to apply the Allais paradox.One prominent theory that distorts the probabilities to this end is the rank-dependent expected utility theory.The new distorted cumulative distribution function (CDF) is given as  ∘  , and then the resulting value function is given as: equation 80 (|) = � ()�()�

Fig. 1 .
Fig. 1.The values of  and  that minimise mean squared error (MSE) Source: author's own calculation.
= 1 when  =   − (  ), then  =  1 .22Subset S of a topological space  is compact if for every open cover of S there exists a finite subcover of S. If  = {  :  ∈ } is an indexed family of sets   , then  is a cover of  if  ⊆ ⋃ ∈   .
(  )} ≡ � ()   () ℝ In the previous expression,    is a PDF of the objective.The certainty equivalent of an allocation is the risk-free amount of money that would make the investor satisfied as a risky allocation equation 124  ↦ () ≡ − ln �   �   ��

Fig. 2
Fig.2 (a and b).The erf utility function of the investor is not a concave function of allocation, and the certainty equivalent for the power utility function is homogeneous Source: author's own calculation.

Table 1 .
The values of α and γ that minimise mean squared error (MSE)