本章导读 本章讲 Kahneman–Tversky (1979) 的前景理论 (Prospect Theory, PT) 及参考点依赖偏好。§2.1 理论：§2.1.1 对期望效用理论的批评——凹效用的 EU 只能解释对大额赌局的风险厌恶，无法解释对小额赌局的风险厌恶（Rabin 2000 校准定理：若拒绝一个小额 50-50 赌局，凹 EU 会荒谬地推出拒绝天量赌局）；§2.1.2 前景理论：编辑阶段 (editing) 六个操作 + 评估阶段 (evaluation) 用价值函数 \(v\) 与权重函数 \(\pi\) 评估前景，给出评估公式 (2.2)、(2.3)；§2.1.3 价值函数：参考点依赖、增益区凹/损失区凸 (\(v''<0\) for \(x>0\), \(v''>0\) for \(x<0\))、损失厌恶导致参考点处有拐点 (kink)（\(v'(x)权重函数：次可加 \(\pi(rp)>r\pi(p)\)（高估小概率）、次确定性 \(\pi(p)+\pi(1-p)<1\)、次比例性 (2.7)，端点跳变，"四重模式"风险态度（图 2.2）。§2.2 为前景理论偏好辩护——好论文的三个特征 + 七个实证证据：2.2.1 一次性购物袋 Homonoff (2018)、2.2.2 高尔夫球手 Pope & Schweitzer (2011)、2.2.3 房市卖家 Genesove & Mayer (2001)、2.2.4 出租车司机 Camerer et al. (1997)、2.2.5 马拉松跑者 Allen et al. (2016)、2.2.6 禀赋效应 Kahneman et al. (1990)、2.2.7 报税 Engström et al. (2015)。§2.3 前景理论的含义（描述性而非规范性）：重构 (reframing)、管理预期、聚合损失（盈余管理，图 2.8）。图 2.1–2.8 均已转述。

三种批评 / Three critiques 期望效用 (EU) 的传统观点：人的偏好可由作用于结果的效用函数 \(u\) 表示。(1) 实验批评（基于实验证据）：Kahneman & Tversky (1979) 的实验室结果清楚表明，人们的偏好并不符合传统 EU 理论。(2) 解析批评（Rabin 2000 校准定理）：EU 用 \(u\) 的凹性来解释风险厌恶，但凹性预测对小额赌局也应极度风险厌恶——这会推出对大额赌局荒谬的风险厌恶。例如：若一个 CRRA 主体拒绝一个"赢 USD 12 / 输 USD 10"的 50-50 赌局，那么由于在长度约 20 的小区间上效用几乎线性，要拒绝该小赌局就需极端风险厌恶，以至于他也会拒绝"赢 USD 1 万亿 / 输 USD 100"的 50-50 赌局——这显然荒谬。故 EU 仅靠凹性能解释大额风险厌恶，却无法解释小额、中额赌局的风险厌恶。(3) 田野实践批评：现实中人们对小额损失表现出强烈风险厌恶——为小家电买保修、只要能买就为小额损失买保险。如此强的小额损失风险厌恶不可能由传统 EU 解释（否则需要荒谬高的曲率），故需要新模型——前景理论（之所以叫"前景"，是因为它研究主体在面对不同"前景/赌局" (prospects) 时如何评估与决策）。The traditional EU view: a person's preferences can be represented by a utility function \(u\) acting on outcomes. (1) Experimental critique (from experimental evidence): the lab results of Kahneman & Tversky (1979) clearly demonstrate that people's preferences are not represented by traditional expected utility theory. (2) Analytical critique (the Rabin 2000 calibration theorem): EU uses the concavity of \(u\) to explain risk aversion, but concavity predicts that risk aversion to small gambles would turn into absurdly large-scale risk aversion. For example, if a CRRA agent turns down a 50-50 gamble of "winning USD 12 / losing USD 10", then since the utility is almost linear over a small interval of length around 20, turning it down requires being so extremely risk averse that he would also turn down a 50-50 gamble of "winning USD 1 trillion / losing USD 100" — which is insane. So EU uses concavity only to explain large-scale risk aversion but can never explain risk aversion to small and moderate gambles. (3) Field-practice critique: in reality people display strong risk aversion to small losses — they buy warranties for small household appliances and buy insurance for small losses whenever they can. Such a strong degree of risk aversion to small losses certainly cannot be explained by traditional EU (otherwise it would require an insanely high curvature), so we need a new model — prospect theory (it is called prospect theory because it is about how agents evaluate and decide when facing different prospects/gambles).

两个阶段 / The two phases 编辑阶段 (Editing phase)：对可得前景做初步分析，得到一个简化表示，包含六个操作：编码 (Coding)——把最终结果转化为损失与增益的域（相对参考点）；合并 (Combination)——把相同结果合并为一个并把概率相加；分离 (Segregation)——把所有前景共享的无风险部分从风险部分中分离出来；取消 (Cancellation)——决策者在评估时丢弃各前景共享的无风险部分；简化 (Simplification)——倾向把概率或结果取整简化；优势识别 (Detection of dominance)——扫描所有前景，发现被占优的前景就直接丢弃、不再深思。评估阶段 (Evaluation phase)：评估已编辑的前景，选取价值最高者。一个已编辑前景的价值 \(V\) 是 \(v\) 与 \(\pi\) 的函数：\(v\) 是价值函数 (value function)，对每个结果 \(x\) 赋一主观价值（定义在前景上、衡量相对参考点的偏离即增益/损失，故 \(v(0)=0\)）；\(\pi\) 是权重函数 (weighting function) \(\pi:[0,1]\to[0,1]\)，把物理概率 \(p\) 映到决策权重 \(\pi(p)\)，在端点收敛于物理概率即 \(\pi(0)=0,\pi(1)=1\)。Editing phase: a preliminary analysis of the available prospects yielding a simplified representation, including six components: Coding — transform final outcomes into domains of losses and gains (relative to a reference point); Combination — same outcomes are combined into one and their probabilities added; Segregation — riskless components shared by all prospects are segregated from the risky components; Cancellation — riskless components shared by all prospects are discarded by the decision maker in evaluation; Simplification — decision makers tend to simplify by rounding the probabilities or outcomes; Detection of dominance — scan all offered prospects and, upon detecting dominated ones, discard them without further thinking. Evaluation phase: edited prospects are evaluated and the one with the highest value is selected. The value \(V\) of an edited prospect is a function of \(v\) and \(\pi\): \(v\) is the value function, assigning a subjective value to each outcome \(x\) (defined on the prospect, measuring deviations — gains/losses — from a reference point, so \(v(0)=0\)); \(\pi\) is the weighting function \(\pi:[0,1]\to[0,1]\), mapping a physical probability \(p\) to a decision weight \(\pi(p)\), converging to the physical probability at the extremes, i.e. \(\pi(0)=0,\pi(1)=1\).

前景与评估规则 / Prospect and evaluation rules 考虑一个简单前景 \((x,p;y,q)\) (2.1)，其中 \(x,y\) 为非零结果、\(p,q\) 为对应物理概率且 \(p+q\le1\)：即以概率 \(p\) 得 \(x\)、概率 \(q\) 得 \(y\)、概率 \(1-p-q\) 得 \(0\)。严格正前景：\(x,y>0\) 且 \(p+q=1\)；严格负前景：\(x,y<0\) 且 \(p+q=1\)；常规前景 (regular)：既非严格正也非严格负。评估规则：对常规前景（\(p+q<1\)，或 \(x\ge0\ge y\)、\(y\ge0\ge x\)）Consider a simple prospect \((x,p;y,q)\) (2.1), where \(x,y\) are non-zero outcomes, \(p,q\) are the associated physical probabilities with \(p+q\le1\): it gives \(x\) with probability \(p\), \(y\) with probability \(q\), and \(0\) with probability \(1-p-q\). Strictly positive prospects: \(x,y>0\) and \(p+q=1\); strictly negative prospects: \(x,y<0\) and \(p+q=1\); regular prospects: neither strictly positive nor strictly negative. Evaluation rule: for regular prospects (either \(p+q<1\), or \(x\ge0\ge y\) or \(y\ge0\ge x\)),

$$V(x,p;y,q)=\pi(p)v(x)+\pi(q)v(y)\tag{2.2}$$

其中 \(v(0)=0,\pi(0)=0,\pi(1)=1\)。此规则形似 EU，只是权重为主观权重而非物理概率。对严格正/负前景（\(p+q=1\)，且 \(x>y>0\) 或 \(xwhere \(v(0)=0,\pi(0)=0,\pi(1)=1\). This rule looks similar to expected utility theory, except the weights are subjective weighting numbers rather than physical probabilities. For strictly positive/negative prospects (\(p+q=1\), and \(x>y>0\) or \(x

$$V(x,p;y,q)=v(y)+\pi(p)\,[v(x)-v(y)]\tag{2.3}$$

这一规则对共享的无风险部分 \(y\) 做了分离。(2.3) 可重排为 \(V=\pi(p)v(x)+[1-\pi(p)]v(y)\)（用 \(p+q=1\)），它仅当 \(\pi(q)+\pi(1-q)=1\) 才退化为 (2.2)，而这一般不成立（见 §2.1.4）。This rule involves segregation of the shared riskless part \(y\). (2.3) can be rearranged to \(V=\pi(p)v(x)+[1-\pi(p)]v(y)\) (using \(p+q=1\)), which degenerates to (2.2) only if \(\pi(q)+\pi(1-q)=1\), which is generally not true (see §2.1.4).

图 2.1（价值函数，已转述 / Figure 2.1, Value Function, paraphrased） 横轴为相对参考点的结果 \(x\)（参考点 = 0），纵轴为价值 \(v(x)\)。右上方（增益域，绿色）是一条凹的递增曲线；左下方（损失域，红色）是一条凸的递增曲线；两段在原点（参考点）相接，损失段斜率明显更陡，整体呈拐折、不对称的 S 形。关键观察：效用对相对参考点的增益与损失作用不同；主体在增益区风险厌恶、在损失区风险偏好；主体损失厌恶（损失处效用下降快于等额增益处效用上升），故曲线在参考点处带拐点且不对称。关键要素：窄框定 (Narrow Framing)——人们不跨赌局聚合，而只盯着当前这个有自己参考点的赌局；损失厌恶 (Loss Aversion)——相对参考点损失同额金钱带来的痛苦大于赢得它带来的快乐，即参考点处斜率的变化（红线比绿线陡）。参考点的可能选择（Kahneman & Tversky 1979）：现状 / 当前资产、期望或抱负水平。The horizontal axis is the outcome \(x\) relative to the reference point (reference point = 0) and the vertical axis is the value \(v(x)\). The upper right (gain domain, green) is a concave increasing curve; the lower left (loss domain, red) is a convex increasing curve; the two meet at the origin (reference point), with the loss segment noticeably steeper, giving an overall kinked, asymmetric S-shape. Key observations: utility applies differently to gains and losses relative to a reference point; agents are risk averse in gains and risk loving in losses; agents are loss averse (utility drops faster in losses than it increases in gains), so the curve is kinked and asymmetric at the reference point. Key ingredients: Narrow Framing — people don't aggregate across gambles, instead they narrowly focus on the current gamble that has its own reference point; Loss Aversion — losing money relative to the reference point hurts more than the happiness gained from the same amount, i.e. the change in slope at the reference point (the red line is steeper than the green). Possible choices for the reference point (Kahneman & Tversky 1979): status quo / current asset, expectation, or aspiration level.

注 2.1 / Remark 2.1 图 2.1 的前景理论偏好能解释对小额与中额赌局的风险厌恶：主体用各自的参考点分别评估每个赌局，而围绕参考点的整体形状（红 + 绿曲线）"足够凹"（因损失厌恶、即参考点处斜率变化），从而能在期望效用性质下于参考点附近生成风险厌恶。The prospect-theory preferences in Figure 2.1 can explain risk aversion to small and moderate scale gambles because agents evaluate each gamble separately with their own reference point, around which the overall shape (red plus green curves) is "concave" enough (because of loss aversion, i.e. the change in slope at the reference point) around their reference point to generate the risk aversion under the expected-utility property.

由"优势识别"推出 \(\pi\) 在中段近似线性 / Detection of dominance ⟹ \(\pi\) essentially linear (derivation) 在编辑阶段假定有"优势识别"过程，则 \(\pi(p)\) 在中段本质上是线性的。设 \(x>y>0\)、\(p>p'\) 且 \(p+q=p'+q'<1\) (2.8)。考虑两前景 \((x,p;y,q)\) 占优 \((x,p';y,q')\)，由 (2.2)：\(\pi(p)v(x)+\pi(q)v(y)>\pi(p')v(x)+\pi(q')v(y)\)，整理得 \([\pi(p)-\pi(p')]v(x)>[\pi(q')-\pi(q)]v(y)\)，即 \(\frac{\pi(p)-\pi(p')}{\pi(q')-\pi(q)}>\frac{v(y)}{v(x)}\) (2.9)。当 \(y\uparrow x\) 时两前景几乎等价，(2.9) 趋于 \(\frac{\pi(p)-\pi(p')}{\pi(q')-\pi(q)}\to1\) (2.10)；又由 (2.8) 有 \(p-p'=q'-q\)，故 \(\frac{[\pi(p)-\pi(p')]/(p-p')}{[\pi(q')-\pi(q)]/(q'-q)}\to1\)，即 \(\pi(\cdot)\) 在不同位置斜率几乎相同——本质上线性。\(\blacksquare\)Assuming a detection-of-dominance process in the editing phase, \(\pi(p)\) is essentially linear in the middle range. Suppose \(x>y>0\), \(p>p'\) and \(p+q=p'+q'<1\) (2.8). Consider two prospects, \((x,p;y,q)\) dominating \((x,p';y,q')\); by (2.2): \(\pi(p)v(x)+\pi(q)v(y)>\pi(p')v(x)+\pi(q')v(y)\), which rearranges to \([\pi(p)-\pi(p')]v(x)>[\pi(q')-\pi(q)]v(y)\), i.e. \(\frac{\pi(p)-\pi(p')}{\pi(q')-\pi(q)}>\frac{v(y)}{v(x)}\) (2.9). When \(y\uparrow x\) the two prospects are almost equivalent, so (2.9) tends to \(\frac{\pi(p)-\pi(p')}{\pi(q')-\pi(q)}\to1\) (2.10); and by (2.8) \(p-p'=q'-q\), so \(\frac{[\pi(p)-\pi(p')]/(p-p')}{[\pi(q')-\pi(q)]/(q'-q)}\to1\), i.e. the slope of \(\pi(\cdot)\) is almost the same at different places — essentially linear. \(\blacksquare\)

图 2.2（权重函数，已转述 / Figure 2.2, Weighting Function, paraphrased） 横轴为物理概率 \(p\)、纵轴为决策权重 \(\pi(p)\)。\(45^\circ\) 对角线表示物理概率（\(\pi=p\)）。\(\pi(p)\) 曲线（红）在小概率处位于对角线上方（高估小概率），在大部分中高概率处位于对角线下方（低估大概率），并在端点 \(p=0\) 与 \(p=1\) 处不"良好"——会突兀地跳到 \(0\) 与 \(1\)。由该权重方案：小概率事件被高估，故投资者可能对低概率增益风险偏好、对低概率损失风险厌恶（因为他们高估了低概率）。权重函数与价值函数共同产生风险态度的"四重模式" (fourfold pattern)：(1) 中等概率增益—风险厌恶；(2) 中等概率损失—风险偏好；(3) 低概率增益—风险偏好；(4) 低概率损失—风险厌恶。The horizontal axis is the physical probability \(p\) and the vertical axis is the decision weight \(\pi(p)\). The \(45^\circ\) diagonal represents physical probability (\(\pi=p\)). The \(\pi(p)\) curve (red) lies above the diagonal at small probabilities (overweighting small probabilities) and below the diagonal over most moderate-to-high probabilities (underweighting large probabilities), and is not well-behaved at the endpoints \(p=0\) and \(p=1\), where it jumps abruptly to \(0\) and \(1\). Under this weighting scheme small-probability events are overweighted, so investors might be risk-seeking over low-probability gains and risk-averse over low-probability losses (because they misperceive the low probability as much higher). The weighting function and value function together generate the fourfold pattern of risk attitudes: (1) risk-aversion over moderate-probability gains; (2) risk-seeking over moderate-probability losses; (3) risk-seeking over low-probability gains; (4) risk-aversion over low-probability losses.

Homonoff (2018)：购物袋税 vs 补贴 / Homonoff (2018): bag tax vs subsidy 自然实验：Arlington County (VA)、Washington D.C.、Montgomery County (MD) 三地相邻、可比。2012/1/1 前 Montgomery 许多商店给"不用一次性袋"的顾客 5 美分返利 (credit)；2012/1/1 后 Montgomery 改为对"用一次性袋"征 5 美分税 (tax)。D.C. 在前后均有袋税；Arlington 在前后均无袋税。参考点 = 每位顾客在加上袋补贴或税之前的结账总价。结果（图 2.3）：Montgomery 的一次性袋使用率在袋税开征后从 Arlington 的水平骤降到 D.C. 的水平。标准经济学不会预期这种跳变，因为 5 美分补贴与 5 美分税对顾客在金钱上完全等价；但结果清楚表明顾客不在乎补贴、却在乎税——这为损失厌恶提供了证据。Natural experiment: Arlington County (VA), Washington D.C., and Montgomery County (MD) are close and comparable. Before 2012/1/1, many stores in Montgomery gave customers a 5-cent credit for not using disposable bags; after 2012/1/1, Montgomery imposed a 5-cent tax for using disposable bags. D.C. had a bag tax both before and after; Arlington had no bag tax both before and after. The reference point is each customer's check-out total price before applying the bag subsidy or tax. Result (Figure 2.3): disposable-bag usage in Montgomery dropped sharply, from the Arlington level to the Washington D.C. level, once the bag tax started. Standard economics would not expect this jump since a 5-cent subsidy and a 5-cent tax are the same thing for customers monetarily; however, the results clearly show consumers don't care about the subsidy but do care about the tax, which provides evidence for loss aversion.

图 2.3（一次性袋使用，已转述 / Figure 2.3, paraphrased） 柱状图"使用一次性袋顾客比例"，按 Arlington（从不征税）、Montgomery（后期征税）、D.C.（始终征税）三地，各画"前期 (pre-period)"与"后期 (post-period)"两根柱。Arlington 两期都约 0.8 且基本不变；D.C. 两期都约 0.35–0.4；Montgomery 前期约 0.8、后期骤降到约 0.45——开征袋税后从 Arlington 水平跌向 D.C. 水平。A bar chart of "proportion of customers using a disposable bag", with two bars ("pre-period" and "post-period") for each of Arlington (never taxed), Montgomery (taxed in the post period), and D.C. (always taxed). Arlington is about 0.8 in both periods and essentially unchanged; D.C. is about 0.35–0.4 in both; Montgomery is about 0.8 pre-period and drops sharply to about 0.45 post-period — falling from the Arlington level toward the D.C. level once the bag tax started.

结果不完全干净：替代解释 / Caveats: alternative explanations 参考点的故事虽合理，但结果并不完全干净。显著性 (salience) 也能解释：补贴时收银员不主动问"要不要袋"，而征税时收银员必须问，从而迫使顾客更注意这一问题、可能因此少用袋。社会规范 (social norm) 也能解释：补贴时不用袋感觉不到"好"，而征税时顾客意识到用袋不被社会认可、要袋时会尴尬，从而少用。The reference-point story is reasonable, but the results are not perfectly clean. Salience could also explain it: with the subsidy the cashier didn't ask whether you need a bag, but with the tax the cashier had to ask, forcing consumers to pay more attention to the question and potentially use fewer bags. Social norm could also explain it: with the subsidy not using bags didn't feel good, but with the tax consumers realized using bags was not favored by society and felt embarrassed demanding bags, so they used fewer.

Pope & Schweitzer (2011)：以"标准杆"为参考点 / par as the reference point 用 PGA 巡回赛 ShotLink 数据（2004–2008、204 场、188 名球手、>500 万次推杆、共 160 万次推杆）来检验损失厌恶。明显的参考点是每洞的标准杆 (par)（脚注：作者也用历史平均作参考点，结果依然显著）。高尔夫中 par 是某洞应有的预定杆数；以"比 par 少 2 杆到多 2 杆"命名为：老鹰 (eagle)、小鸟 (birdie)、标准杆 (par)、柏忌 (bogey)、双柏忌 (double bogey)。推 birdie：若这一杆成功进洞则得 birdie，若没进则最好得 par；推 par：若这一杆成功进洞则得 par，若没进则最好得 bogey。标准经济学认为球手对"推 birdie"与"推 par"应等同看待，因为决定结果的是总杆数而非单洞结果。但前景理论的损失厌恶（以图 2.4 的价值函数）表明：球手更在乎"错过 par 的损失"，而非"打出 birdie 的增益"。预测：球手会为"推 par"投入更多努力与专注，从而"推 par"的成功率高于"推 birdie"。结果（图 2.5）：预测被证实——"推 par"确实比"推 birdie"更可能成功；回归结果亦确认这一发现。Uses PGA Tour ShotLink data (2004–2008, 204 tournaments, 188 golfers, more than 5,000 putts, 1.6 million putts in total) to show loss aversion. The obvious reference point is the par for each hole (footnote: the authors also use the historical average as the reference point, and the results remain significant). In golf, par is the predetermined number of strokes a golfer should complete a hole in; from finishing 2 strokes less to 2 strokes more than par, they are called eagle, birdie, par, bogey, and double bogey. Putt for birdie: the stroke that leads to a birdie if it goes in, with par as the best score otherwise; putt for par: the stroke that leads to par if it goes in, with bogey as the best score otherwise. Standard economics suggests golfers think equivalently about the putt for birdie and the putt for par, because it is the total number of strokes that determines the outcome, not the single-hole result. But prospect-theory loss aversion (with the value function in Figure 2.4) suggests golfers care more about the loss from missing par than about the gain from making a birdie. Prediction: golfers exert more effort and discretion for the putt for par than for birdie, leading to a higher probability of success for the putt for par. Result (Figure 2.5): the prediction is shown to be true — the putt for par is more likely to succeed than the putt for birdie; regression results also confirm this.

图 2.4、2.5（已转述 / Figures 2.4, 2.5, paraphrased） 图 2.4（以 par 为参考点的球手价值函数）：横轴为相对 par 的成绩 \(\Delta s\)（左侧为 bogey/double/triple 等损失、右侧为 birdie/eagle 等增益），纵轴价值 \(V(\Delta s)\)，呈在 par（参考点）处带拐点的 S 形；损失侧（错过 par）斜率更陡，故"从 birdie 掉到 par"的增益小于"从 par 掉到 bogey"的损失——球手更在意保住 par。图 2.5（推 par 与推 birdie 的成功概率）：横轴为到洞距离（英寸），纵轴为进球比例；两条曲线随距离下降，"推 par"曲线在各距离上都略高于"推 birdie"曲线。作者还控制了若干替代解释以使结果更干净可信：球手能力异质性、从此前推杆中学习、洞间差异、控制距离后球在果岭上的位置差异、锦标赛排名差异。Figure 2.4 (value function of golfers with par as the reference point): the horizontal axis is the score relative to par \(\Delta s\) (losses such as bogey/double/triple on the left, gains such as birdie/eagle on the right), the vertical axis is the value \(V(\Delta s)\), an S-shape kinked at par (the reference point); the loss side (missing par) is steeper, so the gain from "birdie down to par" is smaller than the loss from "par down to bogey" — golfers care more about preserving par. Figure 2.5 (probability of success for putt for par vs. for birdie): the horizontal axis is the distance to the hole (inches), the vertical axis is the fraction of putts made; both curves fall with distance, and the "putt for par" curve is slightly above the "putt for birdie" curve at every distance. The authors also control for several alternative explanations to make the results cleaner and more convincing: heterogeneity in player ability, learning from previous putts, hole-specific differences, differences in where the ball is on the green even after controlling for distance, and differences in tournament standings.

Genesove & Mayer (2001)：以"购买价"为参考点 / purchase price as the reference point 用 1990 年代波士顿市中心的数据，说明损失厌恶决定房市卖家行为。参考点 = 每位卖家的购买价。实证设定（回归）：Uses data from downtown Boston in the 1990s to show that loss aversion determines sellers' behavior in the housing market. The reference point is each house seller's purchasing price. The empirical strategy (regression):

$$\text{List Price}=\alpha+\beta\cdot\text{Actual Market Value}+\delta\cdot\text{Loss}+\gamma\cdot\text{Controls}+\varepsilon$$

其中 \(\text{Loss}=\max\{0,\ \text{Purchase price}-\text{current price}\}\)（买入价与现价之差、从下方截断至 \(0\)）。发现：经历"损失"的房主标出更高的挂牌价——损失增加 10% 会导致挂牌价比基本面价格高约 3.5%；把样本限制在贷款价值比 (LTV) < 0.5 的房子，效果类似（排除"LTV 太高（如 >1）卖不掉"的担忧）；在市场上的时间越长，效果越大；个人房主表现出此行为，而机构投资者没有。where \(\text{Loss}=\max\{0,\ \text{Purchase price}-\text{current price}\}\) (the difference between purchase price and current price, truncated from below at \(0\)). Findings: house owners who experience a loss set a higher listing price — a 10% increase in loss leads to about a 3.5% higher listing price than the fundamental price; restricting the sample to houses whose loan-to-value (LTV) ratio is less than 0.5 gives similar effects (this tackles the concern that sellers might not be able to sell if the LTV ratio is too high, e.g. higher than 1); time on the market is increasing in the loss variable; individual house sellers display this behavior while institutional investors don't.

Camerer et al. (1997)：以"目标日收入"为参考点 / target daily wage as the reference point 用纽约市出租车司机的行车单数据，说明司机存在负的劳动供给弹性。参考点 = 司机想达到的目标日收入（约等于平均日工资）。出租车司机是检验劳动供给弹性的好数据源，因为：日工资按天波动（坏天气或大型活动会因需求上升而抬高当日费率）；司机可自己控制工作时长。基本故事：许多司机有"每天挣到一个目标金额"的心态——达到后就回家休息、未达到则继续工作；于是好日子（日工资高、目标易达）工作时长更短、坏日子（日工资低）工作时长更长——这呈现负的劳动供给弹性。该结果为参考点依赖偏好提供了间接证据：要使劳动供给弹性为负，需要在他们通常停止工作的金额（日平均工资）附近有足够的凹性。这篇论文有大量后续文献，许多论文不同意其结果。Uses data on trip sheets of New York City taxi drivers to show the negative labor supply elasticity of those drivers. The reference point is the target daily income drivers want to hit (approximately the average daily wage). Taxi drivers are a good source of data to test labor-supply elasticity because: the daily wage fluctuates day to day (bad weather or big events raise the daily wage rates because more people need a ride); and drivers can control the hours they work. The basic story: many drivers have the mindset of hitting a target amount of money earned each day — after hitting it they go home to rest, before hitting it they keep working; so during good days (higher daily wage, target easy to hit) they work fewer hours, and during bad days (lower daily wage) they work longer hours to hit the target — this displays a negative labor-supply elasticity. The result provides indirect evidence for reference-dependent preferences: for the labor-supply elasticity to be negative, there needs to be enough concavity around the amount of money they typically stop working at, i.e. the daily average wage. There is a huge follow-up literature on this paper, and many papers disagree with its results.

Allen et al. (2016)：以"目标完赛时间"为参考点 / target finishing time as the reference point 用 1970–2013 年 6,888（注：图为 978 万余名）名跑者的马拉松完赛时间数据。参考点 = 完赛的目标时间（典型为 4 小时、4.5 小时等整数阈值）。发现：跑者非常努力地去达到目标时间（如 4 小时），故样本频数在 4 小时处出现尖峰（图 2.6）。跑者在无法达到 4 小时时感到损失，于是当发现快要超过 4 小时时拼命冲刺——导致很多人恰好在 4 小时附近完赛（源于他们最后的超快冲刺）。图 2.6 及围绕几个阈值时间的努力直接度量表明：确实存在跑者厌恶错过的阈值线，支持参考点与损失厌恶的存在。Uses data on the marathon finishing time of 6,888 runners (note: the figure shows 9,789,093 finishers) from 1970 to 2013. The reference point is the target time to finish the marathon (typically 4 hours, 4.5 hours and other thresholds). Findings: runners try very hard to hit the target time (say 4 hours), so there is a peak in sample frequency at 4 hours (Figure 2.6). Runners feel a loss when they cannot hit 4 hours, so they try very hard when they find it's a bit late to hit 4 hours — leading to many runners finishing just around 4 hours as a result of their super-fast final dashing. Figure 2.6 and other direct measures of effort around several threshold times show that there exist certain threshold lines that runners hate to miss, which supports the existence of a reference point and loss aversion.

图 2.6（马拉松完赛时间分布，已转述 / Figure 2.6, paraphrased） 横轴为完赛时间（以 1 分钟为刻度，从 2:00 到 7:00），纵轴为完赛人数（千人）。分布在 3:30–4:00 之间达到峰值后整体右偏下降；在整数/半整数阈值（尤其 4:00）处可见陡降式的尖峰——略快于 4:00 完赛的人数明显多于略慢于 4:00 的人数，反映跑者冲刺以避免"错过 4 小时"的损失。The horizontal axis is finishing time (in one-minute increments, from 2:00 to 7:00) and the vertical axis is the number of finishers (in thousands). The distribution peaks between 3:30 and 4:00 and then declines with a right skew; at round/half thresholds (especially 4:00) there is a sharp spike with an abrupt drop — noticeably more runners finish just faster than 4:00 than just slower, reflecting runners dashing to avoid the loss of "missing 4 hours".

Kahneman et al. (1990)：禀赋效应 / the endowment effect 在实验室中展示禀赋效应：人们倾向把"分到的东西"看得比"没拥有的同样东西"更值钱。实验：参与者以 50-50 概率随机分到两组；一组被给一个马克杯、再问他们愿以多少钱卖出；另一组什么都不给、再问他们愿以多少钱买入。报告的卖价远高于买价（脚注：卖出保留价中位数约 USD 7，买入保留价中位数低于 USD 3），与标准经济学预测相悖。解释：人们若真拥有该杯就更看重它；可能的解释是"放弃自己拥有的杯"的痛苦更大，这支持损失厌恶。故禀赋效应本质上与前景理论中的损失厌恶是同一回事。Shows the endowment effect in a laboratory experiment: people tend to value the things they are given more than the same things they don't own. Experiment: participants are randomly assigned to two groups on 50-50 probability; one group is given a mug and then asked how much they would sell it for; the other group is given nothing and then asked how much they would buy it for. The reported selling price is much higher than the buying price (footnote: the median selling reservation price is around USD 7 while the median buying reservation price is less than USD 3), which is against the prediction of standard economics. Explanation: people value the mug more if they are actually given it; the possible explanation is that the pain from giving up the mug they own is higher, which supports loss aversion. So the endowment effect is basically the same idea as loss aversion in prospect theory.

Engström et al. (2015)：以"年末余额"为参考点 / year-end balance as the reference point 用 2006 年 360 万瑞典纳税人的数据识别损失厌恶。参考点 = 纳税人在年末缴清税负或领取退税之后所剩的金额。考虑两名收入相同、税负同为 USD 30,000 的纳税人 \(A\) 与 \(B\)：\(A\) 的雇主已预扣 USD 29,000，故 \(A\) 年末还需补缴 USD 1,000（损失）；\(B\) 的雇主已预扣 USD 31,000，故 \(B\) 年末可获退税 USD 1,000（增益）。发现：两人行为显著不同——\(A\) 比 \(B\) 更可能去申报抵扣 (deductions) 以降低税负（图 2.7）。这与标准经济学不符（\(A\)、\(B\) 行为应相同），却支持前景理论的损失厌恶：\(A\) 面对的是损失、\(B\) 面对的是增益，\(A\) 比 \(B\) 更在意，说明损失比等额增益更痛苦。Uses data on 3.6 million Swedish taxpayers in 2006 to identify loss aversion. The reference point is the amount of money the taxpayer has at the end of the year after paying for tax liabilities or claiming a tax refund. Consider two taxpayers \(A\) and \(B\) with the same income and the same tax liability of USD 30,000: \(A\)'s employer has already withheld USD 29,000, so \(A\) is supposed to pay USD 1,000 at the end of the year (a loss); \(B\)'s employer has already withheld USD 31,000, so \(B\) is supposed to get a USD 1,000 refund at the end of the year (a gain). Finding: the two people act significantly differently — \(A\) is much more likely than \(B\) to claim deductions to reduce the tax liability (Figure 2.7). This is inconsistent with standard economics (where \(A\) and \(B\) should act the same) but supports loss aversion in prospect theory: \(A\) faces a loss and \(B\) faces a gain, and \(A\) cares more than \(B\), which shows that a loss is more painful than an equal gain.

图 2.7（申报抵扣概率，已转述 / Figure 2.7, paraphrased） 横轴为"初步差额 (preliminary deficit)"（SEK，从 -3000 到 +3000），纵轴为"申报抵扣的概率"。曲线在差额 \(\le0\)（应退税、即增益侧）几乎平坦且较低（约 0.04）；越过 \(0\) 进入正差额（需补缴、即损失侧）后概率明显上升至约 0.06–0.08——面对损失的纳税人更积极申报抵扣，呈现参考点处的不连续上升。The horizontal axis is the "preliminary deficit" (SEK, from -3000 to +3000) and the vertical axis is the "probability of claiming a deduction". The curve is nearly flat and low (about 0.04) for deficit \(\le0\) (refund due, i.e. the gain side); once it crosses \(0\) into a positive deficit (payment due, i.e. the loss side) the probability rises noticeably to about 0.06–0.08 — taxpayers facing a loss claim deductions more actively, showing a discontinuous increase at the reference point.

重构 (Reframing) / Reframing is important 前景理论告诉我们：人们对增益与损失态度不同；我们应利用这一差异来增强想要施加的影响。要阻止人们做某事，应把信息表述为"做了会损失"而非"不做会损失"；要鼓励人们不要在意某事，应把它表述为"额外的增益"，使人们不那么在意它。例（信用卡定价）：美国运通等信用卡公司曾要求商店对刷卡价与现金价收取相同价格。消费者联盟起诉运通并达成和解：运通让步，允许商店使用不同价格，但坚持商店须把信用卡价作为正常价、现金价作为折扣价。这正是前景理论的应用：把"用信用卡的损失"重构为"用现金的增益"，会使顾客更不在意价差，从而仍愿意刷卡消费。Prospect theory tells us that people have different attitudes towards gains and losses; we should take advantage of this difference to increase the power or impact we want to make. To stop people from doing something, phrase the information as a loss from doing it rather than a gain from not doing it. To encourage people not to care about something, frame it as additional gains so that people don't care about it so much. Example (credit-card pricing): American Express and others required stores to use the same credit-card price as the cash price. Consumer Union sued American Express and they negotiated; the outcome was that American Express made a concession to allow stores to use different prices, but insisted that stores use the credit-card price as the normal price and the cash price as a discount price. This is exactly the application of prospect theory: reframing the loss of using a credit card into gains using cash makes customers care less about the price difference, so they are still willing to use credit cards for their purchases.

管理预期 / Manage people's expectations 既然人们对增益与损失看法不同，公司可以管理顾客的预期，使他们永远不会体验到损失，从而降低投诉率。例：航空公司如今总是把航班时长设得比实际预估更长，使航班几乎从不延误。另例：亚马逊总是把"预计送达日期"设得比实际预估更晚。Since people view gains and losses differently, firms can manage customers' expectations such that they never experience a loss, which helps reduce the complaint rate. Example: airlines nowadays always set the flight time on tickets longer than the actual estimated length so that flights almost never arrive late. Another example: Amazon always sets the "estimated delivery date" later than the actual estimated date.

聚合损失（盈余管理）/ Aggregate losses (earnings management) 为降低损失的负面冲击，可把损失聚合 (aggregate)起来，使聚合后的一个大损失比若干小损失之和更不痛（损失侧凸性：\(v\) 在损失域为凸，故 \(v(-a)+v(-b)盈余管理：Burgstahler 和 Dichev (1997) 发现，企业在报告盈余时倾向于有意避免频繁的小额损失——通过对现金流采用不同的入账时点（利用自由裁量），把损失聚合成一次大的损失。因此，报告盈余的分布在 \(0\) 的左侧出现一个低谷 (trough)（图 2.8）。To reduce the negative impact of losses, we can aggregate losses so that the aggregated loss is less painful than the sum of individual small losses (convexity on the loss side: \(v\) is convex over losses, so \(v(-a)+v(-b)earnings management in accounting: Burgstahler and Dichev (1997) find that when reporting earnings, firms tend to intentionally avoid frequent small losses by exploiting the degree of freedom in counting cash flows by different schedules and aggregating losses into a single big loss. Therefore, the distribution of reported earnings has a trough on the left-hand side of zero (Figure 2.8).

图 2.8（盈余管理，已转述 / Figure 2.8, paraphrased） 报告盈余（按盈余区间 $[-0.25,0.35]$ 左右）的频数直方图：分布在略大于 \(0\)（约 \(0.05\)）处达到峰值，但在 \(0\) 的左侧紧邻处出现明显凹陷（低谷）——本应落在"恰好小亏"区间的企业数量异常偏少，它们要么被推到 \(0\) 右侧（报出小额盈利），要么被聚合进更大的亏损区间，正是盈余管理与损失厌恶/损失侧凸性的体现。A frequency histogram of reported earnings (by earnings interval, roughly $[-0.25,0.35]$): the distribution peaks just above \(0\) (around \(0.05\)) but has a clear dip (trough) immediately to the left of \(0\) — the number of firms that should fall in the "just barely a small loss" interval is abnormally low; they are either pushed to the right of \(0\) (reporting a small profit) or aggregated into a larger loss interval, exactly reflecting earnings management and loss aversion / convexity on the loss side.