Games Played on Networks

==Notes by yinung==
原本獨立的 game 在 linked network 玩會如何呢?
本文介紹了網路位置 (network position) 觀念, 及其對均衡的影響 (Ballester, CalvÛ-Armengol & Zenou (2006) Örst establish the connection between equilibrium action and Bonacich centrality (Bonacich 1987),另見 Yves Zenou. (2015),   “Key players" (344 Kb) , In: Y. Bramoullé, B.W. Rogers and A. Galeotti (Eds.), Oxford Handbook on the Economics of Networks, Oxford: Oxford University Press, forthcoming. Vox column)
Date: 2015-03
By: Yann Bramoullé (AMSE – Aix-Marseille School of Economics – EHESS – École des hautes études en sciences sociales – Centre national de la recherche scientifique (CNRS) – Ecole Centrale Marseille (ECM) – AMU – Aix-Marseille Université)
Rachel Kranton (Duke University, Department of Economics – Duke University (Durham, USA))
This chapter studies games played on fixed networks. These games capture a wide variety of economic settings including local public goods, peer effects, and technology adoption. We establish a common analytical framework to study a wide game class. We unearth new connections between games in the literature and in particular between those with binary actions, like coordination and best-shot games, and those with continuous actions and linear best replies. We review and advance existing results by showing how they tie together within the common framework. We discuss the game-theoretic underpinnings of key notions including Bonacich centrality, maximal independent sets, and the lowest and largest eigenvalue. We study the interplay of individual heterogeneity and the network and we develop a new notion – interdependence – to analyze how a shock to one agent affects the action of another agent. We outline directions for future research.

Cobweb Theorems with Production Lags and Price Forecasting

Daniel Dufresne and Felisa Vázquez-Abad (2013). Cobweb Theorems with Production Lags and Price Forecasting. Economics: The Open-Access, Open-Assessment E-Journal, 7 (2013-23): 1—49.
[下載 PDF: Download (pdf, 800.8 kB)](SSCI 期刊)

==noted by yinung==


生產落後期的例子: 挖礦 => 金屬生產
… An important aspect of mining is the lag between the time the decision to increase or decrease production
is made and the time the decision actually takes effect in the market. It takes several years for a planned new mine to start producing, …


Numerical Examples 1. l=落後期變長>=4, 價格波動變成發散


Numerical Examples 2. m=預期價格期數變長, 價格波動變成愈穩定

註: m =0 是典型 cobweb 理論, 即只用當期價格, 預測下一期價格


Numerical Examples 3. c= s/d =供給彈性/需求彈性, c 供需彈性比愈大, 價格波動愈不穩定


==original abstract==

(yinung 翻譯的)

The classical cobweb theorem is extended to include production lags and price forecasts. Price forecasting based on a longer period has a stabilizing effect on prices. Longer production lags do not necessarily lead to unstable prices; very long lags lead to cycles of constant amplitude. The classical cobweb requires elasticity of demand to be greater than that of supply; this is not necessarily the case in a more general setting. Random shocks are also considered.


* Chiarella (1988)studies a system where expected prices follow adaptive expectations, when the demand curve is linear, while the supply curve is non-linear…[The] paper shows that the system is either (1) stable, (2) unstable but cyclical, or (3) chaotic.

* Chiarella and He (2004). That paper studies the cobweb model from a slightly different point of view (in particular, the supply curve is a specific S-shaped function),

Behavioral game theory: Experiments in strategic interaction

Camerer, Colin. Behavioral game theory: Experiments in strategic interaction. Princeton University Press, 2003.

==notes by yinung==

Camerer  (Caltech ) 的書

===有關 beauty contest===

在這個遊戲中,每一個人要猜一個介於 0 – 100 的數字。
誰猜的數字,最接近所有數字之「平均數 x p 」者獲勝

在文獻中 p = 2/3, 或 0.7  。

理論 (p.210), 以 p=2/3 例 (p=0.7 下之點為 100, 70, 49, 34, 24, 17,12,8, 5.76 , 4.03

如果每一個人的選擇, 是均等分配於 0-100, 則平均數 = 50

因此 level-1 thinking 者應該選 50*2/3 = 33 (one-step of reasoning)

因此 level-2 thinking 者應該選 33*2/3 = 22 (two-step of reasoning)

因此 level-3 thinking 者應該選 22*2/3 = 14.8

… 以此類推

選的數字 (67, 100]  (表示預期其它所有人皆選 100) violate first-order iterated dominance

選 (45,67] 符合 a player obeying one step of dominance, but not two.

選 (29,44] 符合 a player obeying two step of dominance, but not three.

Slonim (2001) 有 new player 進來的實驗:

… It turns out that the experienced players use their experience wisely… they choose higher numbers when the new players arrive, and win almost all the time. But their edge disappears after one period.


最早的實驗 Nagel (1995, AER)

第一篇重覆實驗 Ho, Camerer and Weigelt (1998, AER)

很聰明地將 p<1, 再加上 p>1 的實驗進行比對, 難怪可以上 AER (當然不只如此)

文獻回顧類 Nagel (1999) (yinung: 在一本書中)

  • Ho, Teck-Hua, Colin Camerer, and Keith Weigelt. “Iterated dominance and iterated best response in experimental" p-beauty contests"." The American Economic Review 88.4 (1998): 947-969. 提供的 [PDF]

Prospect theory: An analysis of decision under risk

Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, 263-291. 提供的 [PDF]

==notes by yinung==
近數十年來, 影響最大的文章之一, 修正了期望效用理論的一些不合理的部份。
本文大量使用機率組合之選擇,以實驗 (問受試者) 的選擇,來解釋 EU 的不合理。
==original Abstract==
This paper presents a critique of expected utility theory as a descriptive model of decision making under risk, and develops an alternative model, called prospect theory. Choices among risky prospects exhibit several pervasive effects that are inconsistent with the basic tenets of utility theory. In particular, people underweight outcomes that are merely probable in comparison with outcomes that are obtained with certainty. This tendency, called the certainty effect, contributes to risk aversion in choices involving sure gains and to risk seeking in choices involving sure losses. In addition, people generally discard components that are shared by all prospects under consideration. This tendency, called the isolation effect, leads to inconsistent preferences when the same choice is presented in different forms. An alternative theory of choice is developed, in which value is assigned to gains and losses rather than to final assets and in which probabilities are replaced by decision weights. The value function is normally concave for gains, commonly convex for losses, and is generally steeper for losses than for gains. Decision weights are generally lower than the corresponding probabilities, except in the range of low probabilities. Overweighting of low probabilities may contribute to the attractiveness of both insurance and gambling.

Licensing Strategies in the Presence of Patent Thickets

Lihui Lin (2011) “Licensing Strategies in the Presence of Patent Thickets."Journal of Product Innovation Management, Volume 28, Issue 5, pages 698–725, September 2011. DOI: 10.1111/j.1540-5885.2011.00835.x. link to Wiley;;;

Notes by yinung

本文之理論模型是以 2-stage, backward-induction 的方式來推理,其模型之數學形式,可供參考。

文獻上未討論不同 licensing contracts 對 patent thickets, royalty stacking, double marginalization 之影響。

賽局理論模型; 研究下游廠商向上游尋求 N 個 licenses 授權, 其討論 5 種 licensing contract 如下:

  • Quantity-Based Royalty Licenses

下游每單位 output 授權費 u

  • Downstream firm with no bargaining power
    上游完全決定 u (given 下游只能全盤接受, 在本文中稱之 researvation payoff = 0)
  • Downstream firm with any reservation payoff
    上游只要選一個 u 使下游的 payoffs >= reservation payoffs 即可;故下游 bargaining power 會大,則 u 會愈小
  • Revenue-Based Royalty Licenses (by Goldscheider, 1995)
上游向下游收取 r 比例之收入為授權費
  • Profit-Based Royalty Licenses

the price of the final product is independent of the royalty rate and the distribution of bargaining power.

  • Fixed-Fee Licenses

just like a profit-based royalty license, the price of the downstream product is not distorted by upstream costs.

  • Hybrid Licenses: Royalty plus Fixed Fee

some keywords:

patent thickets

royalty stacking

double marginalization


the introduction of a new product or service often requires many complementary technologies


Many key industries (e.g., biomedical, pharmaceuticals, telecommunications, and information technologies) are characterized by cumulative innovations, where the introduction of a new product or service often requires many complementary technologies. When these technologies are protected by intellectual property rights owned by many firms, patent thickets exist, which researchers have argued may hinder the development of cumulative innovations. Specifically, patent thickets may lead to excessive royalty burdens for potential licensees, which is called “royalty stacking,” and if such costs are passed on to consumers, prices of products based on cumulative technologies will be driven up, dubbed as “double marginalization.” The literature, however, does not address these issues under different forms of licensing contracts.

This article develops a game-theoretic model where a downstream firm seeks to license N patents that read on its product from upstream firms. It discusses a variety of licensing forms widely used in practice and attempts to discover whether royalty stacking and double marginalization occur under these forms of licenses. It also studies the impact of bargaining power between parties. It is found that when patent ownership becomes more fragmented, neither royalty stacking nor double marginalization occurs under profit-based royalty, fixed fee, and hybrid licenses. Such problems occur only under pure quantity-based or pure revenue-based royalty licenses when the downstream firm’s bargaining power is low. It is also shown that no matter how fragmented the ownership structure of patent is, hybrid licenses consisting of a fixed fee and a quantity- or revenue-based royalty rate lead to the same market outcomes as a fully integrated firm that owns all the patents and the downstream market.

This article has interesting implications for both research and practice. First, the results show that even under the same patent ownership structure, different forms of licenses lead to quite different market outcomes. Therefore, it is suggested that firms and policy makers pay more attention to contractual forms of licenses when trying to minimize the negative impact of patent thickets. Second, the extant literature has largely assumed that quantity-based royalties are used, where double marginalization is the most severe. In practice, revenue-based royalties are most common, under which double marginalization is much milder. Third, the results show that patent pools can be most effective in mitigating royalty stacking and double marginalization when quantity-based or revenue-based royalties are the sole or primary payment form, especially when downstream firms have low bargaining power.


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Myopic Loss Aversion and the Equity Premium Puzzle

Shlomo Benartzi and Richard H. Thaler  (1995) “Myopic Loss Aversion and the Equity Premium Puzzle.” Quarterly Journal of Economics, Vol. 110, No. 1 (Feb., 1995), pp. 73-92. ; doi: 10.2307/2118511 ; 提供的 [PDF]

==notes by yinung==

Prospect theory, myopic loss aversion (MLA) 和 equity premium puzzle 的重要文獻

==original Abstract==

The equity premium puzzle refers to the empirical fact that stocks have outperformed bonds over the last century by a surprisingly large margin. We offer a new explanation based on two behavioral concepts. First, investors are assumed to be “loss averse,” meaning that they are distinctly more sensitive to losses than to gains. Second, even long-term investors are assumed to evaluate their portfolios frequently. We dub this combination “myopic loss aversion.” Using simulations, we find that the size of the equity premium is consistent with the previously estimated parameters of prospect theory if investors evaluate their portfolios annually.

Stationary Concepts for Experimental 2×2-Games

Selten, Reinhard; Chmura, Thorsten (2008) “Stationary Concepts for Experimental 2×2-Games." American Economic Review, Volume 98, Number 3, June 2008 , pp. 938-966(29). DOI:; 提供的 [PDF];

==notes by yinung==

此文之觀念, 適用於 長期 或 repeated 賽局之均衡解

mixed equilibrium 可視為是動態均衡解 (stationary concept)

Mixed equilibrium has several interpretations. One interpretation is that of a rational recommendation for a one-shot game. Another  interpretation  looks at mixed equilibrium as a result of evolutionary or learning processes in a situation of frequently repeated play with two populations of randomly matched opponents. One may speak of mixed equilibrium as a behavioral stationary concept.

這篇以 2×2 賽局實驗, 比較5種 stationary concept:

Quantal response equilibrium (Richard D. McKelvey and Thomas R. Palfrey 1995)

策略選擇比例是 exponential form

assumes that players give quantal best  responses  to  the behavior of  the others  (see Section  IB).  In  the exponential form of quantal response equilibrium considered here, the probabilities are proportional to an exponential with the expected payoff times a parameter in the exponent.

Action-sampling  equilibrium

從觀察對手的 7 種 (神奇數字) 可能的策略, 來決定自己的最佳策略

…in  a  stationary  situation,  a player takes a sample of seven observations of the strategies played on the other side, and then optimizes against this sample.

Payoff-sampling equilibrium  (Osborne and Rubinstein 1998)


…envisions a  stationary  situation in which a player takes two samples of equal size, one for each of her pure strategies. She then compares the sum of her payoffs in the two samples and plays the strategy with the higher payoff sum…. The best fitting sample size turns out to be six for each of both samples. The name  “payoff-sampling equilibrium" refers to the sampling of own payoffs for each pure strategy.

Impulse balance  equilibrium

和 prospect theory 觀念結合的均衡

proposed  by Selten  is  based  on  learning direction  theory  (Selten  and  Joachim Buchta 1999)…   is  applicable  to the repeated choice of  the same parameter  in  learning situations  in which  the decision maker receives feedback, not only about the payoff for the choice taken, but also for the payoffs connected to alternative actions.
…. The decision maker is assumed to have a tendency to move in the direction of the impulse….

此觀念有別於 reinforcement learning

… that impulse learning is very different from reinforcement learning.

In reinforcement learning, the payoff obtained for a pure strategy played in the preceding period determines the increase of the probability for this strategy. … 完全取決於自己的報酬 (is  entirely based on observed own payoffs)

In  impulse  learning  it  is not  the payoff  in  the preceding period  that  is of crucial importance. It is the difference between what could have been obtained and what has been received, which moves the behavior in the direction of the higher payoff. … 取決於對手的策略和自己的報酬 (requires feedback on the other player’s choice and the knowledge of the player’s own payoff)

均衡成立時, 期望向上和期望向下 impulses 相同, losses 時 impulses 加倍計算
In  the  stationary distribution, expected upward impulses are equal to expected downward impulses. … losses are counted double in the computation of impulses as in prospect theory (Daniel Kahnemann and Amos Tversky 1979).

===五種 equilibrium 之差異===

The five concepts can be thought of as stationary states of dynamic learning models. Learning models differ with respect to their requirements on prior knowledge of the game and on feedback after each period.

reinforcement learning: Nash, quantal response, pay-off sampling equil. 屬之

其它兩種皆需要更多資訊: 自己的報酬 + 對手之選擇

one needs knowledge of one’s own payoff matrix, as well as feedback on the other player’s choice

==original Abstract:==

Five stationary concepts for completely mixed 2×2-games are experimentally compared: Nash equilibrium, quantal response equilibrium, action-sampling equilibrium, payoff-sampling equilibrium (Martin J. Osborne and Ariel Rubinstein 1998), and impulse balance equilibrium. Experiments on 12 games, 6 constant sum games, and 6 nonconstant sum games were run with 12 independent subject groups for each constant sum game and 6 independent subject groups for each nonconstant sum game. Each independent subject group consisted of four players 1 and four players 2, interacting anonymously over 200 periods with random matching. The comparison of the five theories shows that the order of performance from best to worst is as follows: impulse balance equilibrium, payoff-sampling equilibrium, action-sampling equilibrium, quantal response equilibrium, Nash equilibrium.


引自: Daniel Friedman and Alessandra Cassar, Economics Lab: An Intensive Course in Experimental Economics, Routledge, London and New York (2004) pp. 248


Hertwig, R. and Ortman, A. (2001) “Experimental practices in economics: a challenge for psychologist? ” Behavioral and Brain Sciences, 24:383–403. link to PDF

兩學門間有4 種主要之差異 (p.18)

  1. Script versus open-ended

  2. Repeated trials versus one-shot

  3. Salient pay

  4. Deception

主要差異項目 經濟實驗 心理實驗
實驗說明 script 強調完整之角色定義, 互動模式, 報酬如何決定
Economists (drawing on Siegel’s tradition) almost always include detailed formal written instructions for subjects defining their roles, interactions, and payoffs. Psychologists in recent decades seldom use written instructions and usually are quite casual about describing the task to the subjects.
Psychologists in recent decades seldom use written instructions and usually are quite casual about describing the task to the subjects
實驗進行 受試者重覆試驗相同之情境 repeated trials
Economists since Smith (1962) typically have subjects repeat the task or interactions several times, and focus on data from later repetitions when they are sure that the subjects are fully adjusted to the environment.
受試者只進行一次實驗 one-shot Psychologists more commonly just give the subjects one shot at any particular task.
受試者報酬 以實驗參與之表現優劣支付現金
Economists almost always pay subjects in cash based on performance.
很少支付現金, 且是固定金額
Psychologists seldom pay cash, and when they do, they usually pay a flat amount unrelated to performance.
資訊真實性 幾乎不會採用誤導
Deception of any kind is taboo among experimental economists.
A large fraction of social psychology experiments attempt to mislead the subjects as to the true nature of the task.


Role of theory

Economics has a core theory that assumes self-interest, rationality, and equilibrium. Theory in psychology is more descriptive and eclectic. Hence, psychologists are less concerned with salient rewards, repeated trials, etc. that give economic theory a better shot.

Role of institutions

Following a suggestion by Sunder in Friedman and Sunder (1994), Figure 2.2 presents a spectrum of economic situations based on the institutional constraints. Personal preferences are the dominant influence in individual choice tasks but play a minor role for firms in strong institutions such as markets. Cognitive psychologists prefer to work at the weak institution end of the spectrum, while social psychologists study quite different social constraints. By contrast, economic analysis emphasizes the institutional constraints, as we will see in Chapter 3.

實驗說明 script open-ended
實驗進行 r r

Economics in the Laboratory

Vernon L. Smith (1994) “Economics in the Laboratory." Journal of Economic Perspectives ,Vol. 8, No. 1, Winter, 1994 , pp. 113-131. (link to JSTOR

==Summary by yinung==

Smith 在此文回顧了實證經濟學的重要貢獻 (其題為 “為何經濟學家要做實驗?" (why do economists conduct experiments?)), 他列舉以下 7 大理由,其實就是經濟實驗的主要功能:


1. Test a theory,or discriminate between theories

2. Explore the causes of a theory’sfailure.

3. Establish empirical regularities as a basis for new theory

4. Compare environments.

5. Compare institutions

6. Evaluate policy proposals.

7. The laboratory as a testing ground for institutional design.


Smith 並總結了經濟實驗多年來所發現之重要結論 (基於 制度 (institutions)、環境 (environments)、與行為 (behaviors) 之架構; 不過我認為還要加上「資訊」(information) 和 「預期之形成」(expectations formation) ):

1. Institutions Matter

2. Unconscious Optimization in Market Interactions

3. Information: Less Can Be Better

4. Common Information is Not Sufficient to Yield Common Expectations or “Knowledge"

5. Dominated Strategies Are for Playing, Not Eliminating

6. Efficiency and Underrevelation Are Compatible

7. The Endowment Effect

8. Fairness: Taste or Expectation?


水星運行之軌道未遵循牛頓運動定律之預測,並沒有讓科學家認為該運動定律是錯誤的,反而是斷定水星和太陽中間,必定有一個未知的行星 (甚至被命名為火神星 (Vulcan) [註 1] ) 干擾了水星應該運行之路徑… (Smith, 1994, p.127)

Smith 指出,任何對理論的驗證 (test),都避免不了必要的輔助假設 (auxiliary hypotheses),如此才能將觀察的結果解釋為對該理論的驗證結果。這些「輔助假設」常有很多不同的名詞,諸如: 啟始條件 (initial conditions) 、其它條件相同 (ceteris paribus clauses)、背景資訊 (background information)、… 等。

因此可以說,所有對理論的驗證,都隱含是一種聯合檢定 (joint tests) ; 亦即,對理論的驗證皆是條件式檢定,其皆建立在輔助假設 (auxiliary hypotheses) 成立的前提之下。

這就為何實驗文獻之發現,常被批評的原因, 如同「杜恩─奎因論題」(Duhem-Quine Thesis) 所述之「指任何理論儘管出現推翻它的證據,我們只要在我們的信念網絡中作一點改變,便可以繼續接受該理論」 (see 淺談「杜咸-奎因論旨」與非充分決定性論證)

Smith 在此文並對一些對實驗經濟研究法之批評,提出反駁。例如 Harrison (1992) 認為經濟實驗所得到的錯誤觀是因為受試者的實驗報酬之機會成本太低,所以才會偏離理論之預測。Smith and Walker (1993) 的 近 30 年文獻回顧說明了受試者所獲之「貨幣報酬的確可以改善受試者之表現 (在某些條件之下)」,但其它除了「受試者報酬」之條件也同樣重要,因為增加受試者報酬並沒有讓那些偏離理論所預測的觀察消失 (many anomalies do not disappear by escalating payoffs, p.127)

… 未完成, 待續…


Smith, Vernon L., and James M. Walker, “Monetary Rewards and Decision Cost in Experimental Economics," Economic Inquiry,April 1993, 31, 245-61.

註 1: 水星偏離軌道 (水星的軌道偏離正圓程度很大,近日點距太陽僅四千六百萬千米,遠日點卻有7千萬千米,在軌道的近日點它以十分緩慢的速度按歲差圍繞太陽向前運行, see 背景資料:神秘的水星) 的真正原因,普遍相信可以由爱因斯坦所提出的广义相对论所解釋,即由于太阳质量较大,水星受它的引力作用也较大,在其附近的空间发生了较大的“弯曲” 所致。