1) System reliability can depend on the combined efforts of multiple individuals, making it a public good prone to free riding.
2) There are three prototypical cases for how individual effort relates to outcomes: total effort, weakest link, and best shot.
3) In a total effort scenario, the individual with the highest benefit-cost ratio contributes all the effort while others free ride. In weakest link, there is a range of equilibria with effort varying from zero to the maximum of the lowest benefit-cost agent. In best shot, the highest or lowest benefit-cost agent may contribute all the effort.
Density function in probability or density of any continuous instantly selected variable is a function in which the count provided at given point (sample) in the available set of possibilities random values can be predicted as giving a linked or dependent prospect for a continuous unplanned variable would the same of that sample. Copy the link given below and paste it in new browser window to get more information on Density Function:- www.transtutors.com/homework-help/statistics/density-function.aspx
Density function in probability or density of any continuous instantly selected variable is a function in which the count provided at given point (sample) in the available set of possibilities random values can be predicted as giving a linked or dependent prospect for a continuous unplanned variable would the same of that sample. Copy the link given below and paste it in new browser window to get more information on Density Function:- www.transtutors.com/homework-help/statistics/density-function.aspx
This presentation is an attempt to introduce Game Theory in one session. It's suitable for undergraduates. In practice, it's best used as a taster since only a portion of the material can be covered in an hour - topics can be chosen according to the interests of the class.
The main reference source used was 'Games, Theory and Applications' by L.C.Thomas. Further notes available at: http://bit.ly/nW6ULD
Applications of game theory on event management Sameer Dhurat
Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science and biology.
In this presentation ,discussed regarding Application of game theory on Event Management with the help of Prisoner's Dilemma Game
This presentation about game theory particularly two players zero sum game for under graduate students in engineering program. It is part of operations research subject.
Astra Women’s Business Alliance believes that the young women today are the leaders and visionaries of tomorrow. With that belief that Astra President and Vice-President, Diane McClelland and Suzanne Lackman, started the Astra S.T.E.A.M. Program in 2013. The Astra S.T.E.A.M. Program takes a holistic, pathways approach to S.T.E.A.M. Education. Following a pathways approach, encourages a shift in focus from numbers of people in S.T.E.A.M. jobs and people with STEAM degrees to focusing on equipping individuals, specifically girls, with the skills needed to thrive and grow in the workforce.
Peter De Keyzer, Chief Economist, BNP Paribas Fortis. Presented at Crowdsourcing Week Europe 2015. For more information or to join the next event: http://crowdsourcingweek.com/
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Sindrome di Alström - La relazione di Kay Parkinson AlstromItalia
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By Marie Noelle Keijzer. Presented at Crowdsourcing Week Europe 2016. For more information and details on our next event, visit www.crowdsourcingweek.com.
Ransomware makers have progressed from infecting individual users on a one-off basis to targeting whole organizations. Using advanced attack techniques like reconnaissance and credential abuse, they can infect many machines at once, bringing business operations to a screeching halt.
To stop individual and network-wide attacks, organizations need a multi-layered defense that includes both detection and eradication. Join security experts from LightCyber and Ayehu as they discuss how to defeat ransomware. They will present a live demo of a ransomware attack and show how to quickly contain it with LightCyber + Ayehu.
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- Why targeted ransomware is more devastating and difficult to stop than traditional ransomware.
- Methods to disrupt ransomware attacks and ward off extortionists.
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Presenters:
- Kasey Cross, Sr. Product Manager, LightCyber
- Guy Nadivi, Director, Business Development – North America, Ayehu
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This presentation is an attempt to introduce Game Theory in one session. It's suitable for undergraduates. In practice, it's best used as a taster since only a portion of the material can be covered in an hour - topics can be chosen according to the interests of the class.
The main reference source used was 'Games, Theory and Applications' by L.C.Thomas. Further notes available at: http://bit.ly/nW6ULD
Applications of game theory on event management Sameer Dhurat
Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science and biology.
In this presentation ,discussed regarding Application of game theory on Event Management with the help of Prisoner's Dilemma Game
This presentation about game theory particularly two players zero sum game for under graduate students in engineering program. It is part of operations research subject.
Astra Women’s Business Alliance believes that the young women today are the leaders and visionaries of tomorrow. With that belief that Astra President and Vice-President, Diane McClelland and Suzanne Lackman, started the Astra S.T.E.A.M. Program in 2013. The Astra S.T.E.A.M. Program takes a holistic, pathways approach to S.T.E.A.M. Education. Following a pathways approach, encourages a shift in focus from numbers of people in S.T.E.A.M. jobs and people with STEAM degrees to focusing on equipping individuals, specifically girls, with the skills needed to thrive and grow in the workforce.
Peter De Keyzer, Chief Economist, BNP Paribas Fortis. Presented at Crowdsourcing Week Europe 2015. For more information or to join the next event: http://crowdsourcingweek.com/
Tm CMS Tool | TM CMS Tool development company in bangalore.NS Web Technology
TM Content Management System Tool is an in-house product of NS Web. TM CMS is adaptable & easy-to-handle platform offering end-to-end customization feasibility suiting industry needs and standards.
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Sindrome di Alström - La relazione di Kay Parkinson AlstromItalia
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By Marie Noelle Keijzer. Presented at Crowdsourcing Week Europe 2016. For more information and details on our next event, visit www.crowdsourcingweek.com.
Ransomware makers have progressed from infecting individual users on a one-off basis to targeting whole organizations. Using advanced attack techniques like reconnaissance and credential abuse, they can infect many machines at once, bringing business operations to a screeching halt.
To stop individual and network-wide attacks, organizations need a multi-layered defense that includes both detection and eradication. Join security experts from LightCyber and Ayehu as they discuss how to defeat ransomware. They will present a live demo of a ransomware attack and show how to quickly contain it with LightCyber + Ayehu.
Attendees will learn:
- Why targeted ransomware is more devastating and difficult to stop than traditional ransomware.
- Methods to disrupt ransomware attacks and ward off extortionists.
- How LightCyber + Ayehu work together to detect and mitigate ransomware threats.
Presenters:
- Kasey Cross, Sr. Product Manager, LightCyber
- Guy Nadivi, Director, Business Development – North America, Ayehu
- Peter Lee, Director, Professional Services, Ayehu
Julian Petrin, Founder, Nexthamburg. Presented at Crowdsourcing Week Europe 2015. For more information or to join the next event: http://crowdsourcingweek.com/
Watch this presentation and learn all about Microservices.
*Flannel, Weave, IPVLAN, MacVLAN and how they fit together with Docker, Swarm or Kubernetes
*How containers communicate with each other
*How the choice of Networking Interface impacts router and switch deployment in the Data Center
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Chapter 3. Arbitrage (asset) pricing theory
Neil Wallace
December 26, 2017
1 Introduction
Arbitrage possibilities arise when there are different transactions that achieve
the same final outcome. When there are such possibilities, sharp conclusion
can be drawn about the prices in the different transactions. Consider again
the two-state rainfall model and assume that in addition to making trades
contingent on the outcome of the rainfall, each person can buy or sell plots
of land before rainfall is realized. There are N plots of land and the owner
of plot n gets (wn1, wn2); that is, wn1 is the rice crop on land-n if rainfall is
low and wn2 is that crop if rainfall is high. If wn1 6= wn2, then land-n is what
would ordinarily be called a risky asset. We can use arbitrage to compute
the price of a given plot land in terms of the prices of rice contingent on the
rainfall outcome, the prices we defined above. The general version of doing
that is called arbitrage-pricing theory (or APT).
Suppose p = (p1, p2) is the price of outcome contingent rice. The APT
says that the price-of-land n is p1wn1+ p2wn2. In order to reach that conclu-
sion, assume that any person can either buy land-n or go-short land-n. (To
go-short land-n means promising to pay out wn1 amount of rice if rainfall is
high and wn2 if rainfall is low.) Let v(wn1, wn2) be the pre-outcome price of
land-n (in abstract units of account).
Exercise 1 Suppose contingent rice can be bought or sold at p = (p1, p2).
What purchases and sales would be profitable (without bearing risk) if v(wn1, wn2)
< p1wn1 + p2wn2? What purchases and sales would be profitable (without
bearing risk) if v(wn1, wn2) > p1wn1 + p2wn2? (Hint: Answer this question
by assuming, that payment for a plot of land is made by making the follow-
ing promise: If v(wn1, wn2) is the price of land (wn1, wn2), then it can be
purchased for any (x1, x2) that satisfies v(wn1, wn2) = p1x1 + p2x2.)
1
Now let’s leave rice, land, and two states behind, and exposit APT more
generally. We assume that there are S states and that p = (p1, p2, ..., pS)
denotes the price vector of outcome-contingent goods. We define assets by
their payoffs. In particular, an asset is a vector of outcome-specific payoffs–
say, (a1, a2, ..., aS) with the interpretation that the owner of this asset receives
as units of the good if outcome s occurs. (If as < 0, then the owner must
pay out as units of the good if outcome s occurs.) The price of the asset is
the pre-outcome price at which the asset can be bought or sold. We denote
that price v(a1, a2, ..., aS).
Proposition 1 The only price of the asset consistent with no arbitrage prof-
its is
v(a1, a2, ..., aS) =
S∑
s=1
psas. (1)
Exercise 2 Argue that proposition 1 is true. (Hint: Show that each person
would profit by buying the asset if v(a1, a2, ..., aS) <
∑S
s=1 psas and that no-
one wants to buy it; then show that each person would profit by selling it
short if v(a1, a2, ..., aS) >
∑S
s=1 psas.)
Co ...
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Agent-based economic modeling often requires the determination of an initial equilibrium price vector. Calculating this directly requires algorithms of exponential computational complexity. It is known that a partial equilibrium price can be estimated using a median of trades. This paper explores the possibility of a multivariate generalization of this technique using depth functions.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Biological screening of herbal drugs: Introduction and Need for
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Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Model Attribute Check Company Auto PropertyCeline George
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Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...
Reliability
1. System Reliability and Free Riding
Hal R. Varian∗
University of California, Berkeley
February 2001
This version: November 30, 2004.
System reliability often depends on the effort of many individuals, making reli-
ability a public good. It is well-known that purely voluntary provision of public
goods may result in a free rider problem: individuals may tend to shirk, resulting
in an inefficient level of the public good.
How much effort each individual exerts will depend on his own benefits
and costs, the efforts exerted by the other individuals, and the technology that
relates individual effort to outcomes. In the context of system reliability, we
can distinguish three prototypical cases.
Total effort. Reliability depends on the sum of the efforts exerted by the in-
dividuals.
Weakest link. Reliability depends on the minimum effort.
Best shot. Reliability depends on the maximum effort.
Each of these is a reasonable technology in different circumstances. Suppose
that there is one wall defending a city and the probability of successful defense
depends on the strength of the wall, which in turn depends on the sum of the
efforts of the builders. Alternatively, think of the wall as having varying height,
with the probability of success depending on the height at its lowest point. Or,
finally, think of a there being several walls, where only the highest one matters.
Of course, many systems involve a mixture of these cases.
1 Literature
Hirshleifer [1983] examined how public good provision varied with the three
technologies described above. His main results were:
∗ Thanks to Steve Shavell for useful suggestions about liability rules. Research support
from NSF grant SBR-9979852 is gratefully acknowledged.
1
2. 1. With the weakest-link technology, there will be a range of Nash equilibria
with equal contributions varying from zero to some maximum, which is
determined by the tastes of one of the agents.
2. The degree of under provision of the public good rises as the number of
contributors increases in the total effort case, but the efficient amount of
the public good and the Nash equilibrium amount will be more-or-less
constant as the number of contributors increases.
3. Efficient provision in the best-effort technology generally involves only the
agents with the lowest cost of contributing making any contributions at
all.
Cornes [1993] builds on Hirshleifer’s analysis. In particular he examines the
impact of changes in income distribution on the equilibrium allocation. Sandler
and Hartley [2001] provide a comprehensive survey of the work on alliances,
starting with the seminal contribution of Olson and Zeckhauser [1966]. Their
motivating concern is international defense with NATO as a recurring exam-
ple. In this context, it is natural to emphasize income effects since countries
with different incomes may share a greater or lesser degree of the burden of an
alliance.
The motivating example for the research reported here is computer system
reliability and security where teams of programmers and system administrators
create systems whose reliability depends on the effort they expend. In this
instance, considerations of costs, benefits, and probability of failure become
paramount, with income effects being a secondary concern. This difference in
focus gives a different flavor to the analysis, although it still retains points of
contact with the earlier work summarized in Sandler and Hartley [2001] and the
other works cited above.
2 Notation
Let xi be the effort exerted by agent i = 1, 2, and let P (F (x1 , x2 )) be the
probability of successful operation of the system. Agent i receives value vi from
the successful operation of the system and effort xi costs the agent ci xi .
The expected payoff to agent i is taken to be
P (F (x1 , x2 ))vi − ci xi
and the social payoff is
P (F (x1 , x2 ))[v1 + v2 ] − c1 x1 − c2 x2 .
We assume that the function P (F ) is differentiable, increasing in F , and is
concave, at least in the relevant region.
We examine three specifications for F , motivated by the taxonomy given
earlier.
2
3. Total effort. F (x1 , x2 ) = x1 + x2 .
Weakest link. F (x1 , x2 ) = min(x1 , x2 ).
Best shot. F (x1 , x2 ) = max(x1 , x2 ).
3 Nash equilibria
We first examine the outcomes where each individual chooses effort unilaterally,
and then compare these outcomes to what would happen if the efforts were
coordinated so as to maximize social benefits minus costs.
3.1 Total effort
Agent 1 chooses x1 to solve
max v1 P (x1 + x2 ) − c1 x1 ,
x1
which has first-order conditions
v1 P (x1 + x2 ) = c1 .
Letting G be the inverse of the derivative of P , we have
x1 + x2 = G(c1 /v1 ).
Defining x1 = G(c1 /v1 ) we have the reaction function of agent 1 to agent 2’s
¯
choice
f1 (x2 ) = x1 − x2 .
¯
Similarly
f2 (x1 ) = x2 − x1 .
¯
These reaction functions are plotted in Figure 1. It can easily be seen that
the unique equilibrium involves only one agent contributing effort, with the
other free riding, except in the degenerate case where each agent has the same
benefit/cost ratio: v2 /c2 = v1 /c1 .
Let us suppose that v2 /c2 > v1 /c1 . Then, x2 > x1 , so agent 2 contributes
¯ ¯
everything and agent 1 free rides.
Fact 1 In the case of total effort, system reliability is determined by the agent
with the highest benefit-cost ratio. All other agents free ride on this agent.
The fact that we get this extreme form of free riding when utility takes
this quasilinear form is well-known; see, for example, Varian [1994] for one
exposition.
3
4. x2
Nash equilibrium
x1
Figure 1: Nash equilibrium in total effort case.
3.2 Weakest link
Agent 1’s problem is now
max v1 P (min(x1 , x2 )) − c1 x1 .
x1
It is not hard to see that agent 1 will want to match agent 2’s effort if x2 < x1 ,
¯
and otherwise set x1 = x1 . The two agents’ reaction functions are therefore
¯
f1 (x2 ) = min(x2 , x1 )
¯ (1)
f2 (x1 ) = min(x1 , x2 ).
¯ (2)
These reaction functions are plotted in Figure 2. Note that there will be a whole
range of Nash equilibria. The largest of these will be at min(¯1 , x2 ). This Nash
x ¯
equilibrium Pareto dominates the others, so it is natural to think of it as the
likely outcome.
Fact 2 In the weakest-link case, system reliability is determined by the agent
with the lowest benefit-cost ratio.
3.3 Best shot
In the best-shot case it is not hard to see that there will always be a Nash equi-
librium where the agent with the highest benefit-cost ratio exerts all the effort.
What is more surprising is that there will sometimes be a Nash equilibrium
where the agent with the lowest benefit-cost ratio exerts all the effort.1 This
can occur when the agent with the highest benefit-cost ratio chooses to exert
zero effort, leaving all responsibility to the other agent.
1I am grateful to Xiaopeng Xu for pointing this out to me.
4
5. f1 (x2 )
x2
f 2 (x1 )
Nash equlibrium
x1
Figure 2: Nash equilibrium in weakest link case.
To see an example of this, suppose that the agents’ utility functions have
the form vi ln x − xi where x = min(x1 , x2 ). (True, ln x is not a probability
distribution, but that makes no difference for what follows.)
The first-order condition is vi /x = 1, so x1 = v1 or 0, depending on whether
v1 ln x1 − x1 is greater or less than v1 ln x2 . Hence x1 = v1 if x2 ≤ v1 /e and
x1 = 0 if x2 ≥ v1 /e.
In order to create a simple example, suppose that v1 = e and v2 = 2e. This
gives us x1 = e for x2 ≤ 1 and zero otherwise, while x2 = 2e for x1 ≤ 2 and
zero otherwise. These reaction curves are depicted in Figure 3. Note that in
the case depicted there are two equilibria, with each agent free-riding in one of
the equilibria. (If the reaction curves were continuous, there would be another
equilibrium, but the discontinuity prevents that.)
In the “slacker equilibrium,” the agent with the highest benefit cost ratio
chooses to contribute zero, knowing that the other agent will be forced to con-
tribute. Given that the other agent does contribute, it is optimal for the slacker
to contribute zero.
The three baseline cases we have studied, total effort, weakest link, and
best shot have three different kinds of pure-strategy Nash equilibria: unique,
continuum, and (possibly) two discrete equilibria.
4 Social optimum
4.1 Total effort
The social problem solves
max P (x1 + x2 )[v1 + v2 ] − c1 x1 − c2 x2 .
x1 ,x2
5
6. x2
8
7
6 Nash equilibrium
5
4
3
2
1
Nash equilibrium
1 2 3 4 5 6 7 8 x1
Figure 3: Nash equilibria in best-shot case.
The first-order conditions
P (x1 + x2 )[v1 + v2 ] ≤ c1 (3)
P (x1 + x2 )[v1 + v2 ] ≤ c2 . (4)
At the optimum, the agent with the lowest cost exerts all the effort. Let cmin =
min{c1 , c2 }, so that the optimum is determined by
x∗ + x∗ = G(cmin /(v1 + v2 )).
1 2 (5)
Summarizing, we have:
Fact 3 In the total effort case, there is always too little effort exerted in the
Nash equilibrium as compared with the optimum. Furthermore, when v2 /c2 >
v1 /c1 but c1 < c2 , the “wrong” agent exerts the effort.
4.2 Best shot
The social and private outcomes in this case are the same as in the total effort
case.
4.3 Weakest link
The social objective is now
max P (min(x1 , x2 ))[v1 + v2 ] − c1 x1 − c2 x2 .
x1 ,x2
At the social optimum, it is obvious that x1 = x2 so we can write this problem
as
max P (x)[v1 + v2 ] − [c1 + c2 ]x,
x
6
7. which has first-order conditions
P (x)[v1 + v2 ] = c1 + c2 ,
or
x1 = x2 = x = G((c1 + c2 )/(v1 + v2 )). (6)
Fact 4 The probability of success in the socially optimal solution is always lower
in the case of weakest link that in the case of total effort.
This occurs because the weakest link case requires equal effort from all the
agents, rather than just effort from any single agent. Hence it is inherently more
costly to increase reliability in this case.
5 Identical values, different costs
Let n be the number of agents and, for simplicity, set vi = 1 for all i = 1, . . . , n.
In the total-effort case, the social optimum is given by
nP (x) = min ci ,
while the private optimum is determined by
P (x) = min ci .
In the weakest-link case, the social optimum is determined by
nP (x) = ci ,
i
or
1
P (x) = c =
¯ ci .
n i
while the private optimum is determined by
P (x) = max ci .
If we think of drawing agents from a distribution, what matters for system
reliability are the order statistics—the highest and lowest costs of effort.
Fact 5 Systems will become increasingly reliable as the number of agents in-
creases in the total efforts case, but increasingly unreliable as the number of
agents increases in the weakest link case.
7
8. 6 Increasing the number of agents
Let us now suppose that vi = ci = 1 and that the number of agents is n. In this
case, the social optimum in the case of total effort is determined by
nP ( xi ) = 1,
i
or
xi = G1/n).
i
The Nash equilibrium satisfies
P ( xi ) = 1,
i
or
xi = G(1).
i
Fact 6 In the total efforts case with identical agents, the Nash outcome remains
constant as the number of agents is increased, but the socially optimal amount
of effort increases.
In weakest-link case, the social optimum is determined by
nP (x) = n,
which means that the socially optimal amount of effort remains constant as n
increases. In the Nash equilibrium
P (x) = 1,
or
x = G(1).
Fact 7 In the weakest-link case with identical agents, the socially optimal reli-
ability and the Nash reliability are identical, regardless of the number of agents.
7 Fines and liability
7.1 Total effort
Let us return to the two-agent case, for ease of exposition, and consider the
optimal fine, that is, the fine that induces the socially optimal levels of effort.
Let us start with the total effort case, and suppose that agent 1 has the lowest
marginal cost of effort. If we impose a cost of v2 on agent 1 in the event that
the system fails, then agent 1 will want to maximize
v1 P (x1 + x2 ) + v2 [1 − P (x1 + x2 )] − c1 x1 .
8
9. The first order condition is
(v1 + v2 )P (x1 + x2 ) = c1 ,
which is precisely the condition for social optimality. This result easily extends
to the n-person case, so we have:
Fact 8 A fine equal to the costs imposed on the other agents should be imposed
on the agent who has the lowest cost of reducing the probability of failure.
Alternatively, we could consider a strict liability rule, in which the amount
charged in the case of system failure is paid to the other agent. If the “fine” is
paid to agent 2, his optimization problem becomes
v2 P (x1 + x2 ) + [1 − P (x1 + x2 )]v2 − c2 x2 .
Simplifying, we have
v 2 − c 2 x2 ,
so agent 2 will want to set x2 = 0. But this is true in the social optimum as well,
so there is no distortion. Obviously this result is somewhat delicate; in a more
general specification, there would be some distortions from the liability payment
since it will, in general, change the behavior of agent 2. If the liability payment
is too large, it may induce agent 2 to seek to be injured. This is not merely
a theoretical issue, as it seems likely that if liability rules would be imposed,
each system failure would give rise to many plaintiffs, each of whom would seek
maximal compensation.
The fact that the agents with the least cost of effort to avoid system failure
should bear all the liability is a standard result in the economic analysis of tort
law, where it is sometimes expressed as the doctrine of the “least-cost avoider.”
As Shavell [1987], page 17-18, points out, this doctrine is correct only in rather
special circumstances, of which one is the sum-of-efforts case we are considering.
7.2 Weakest link
How does this analysis work in the weakest-link case? Since an incremental
increase in reliability requires effort to be exerted by both parties, each agent
must take into account the cost of effort of the other.
One way to do this is to make each agent face the other’s marginal cost, in
addition to facing a fine in case of system failure. Letting x = min{x1 , x2 }, the
objective function for agent 1, say, would then be:
v1 P (x) − [1 − P (x)]v2 − c1 x1 − c2 x1 .
Agent 1 would want to choose x = x1 determined by
(v1 + v2 )P (x) = c1 + c2 ,
9
10. which is the condition for social optimality. Agent 2 would make exactly the
same choice.
Let us now examine a liability rule in which each must compensate the other
in the case of system failure. The objective functions then take the form
maxx1 v1 P (x) − (1 − P (x))v2 + (1 − P (x))v1 − c1 x1 (7)
maxx2 v2 P (x) − (1 − P (x))v1 + (1 − P (x))v2 − c2 x2 (8)
(9)
Note that when the system fails, each agent compensates the other for their
losses, but is in turn compensated.
Simplifying, we can express the optimization problems as
maxx1 v1 − v2 + v2 P (x) − c1 x1 (10)
maxx2 v2 − v1 + v1 P (x) − c2 x2 (11)
This leads to first order conditions
v2 P (x) = c1 (12)
v1 P (x) = c2 (13)
If we are in the symmetric case where v1 = v2 and c1 = c2 (or more generally,
where v1 c1 = v2 c2 ), then both of these equations can be satisfied and, somewhat
surprisingly, the solution is the social optimum. Of course, if all agents are
identical, then there is no reason to impose a liability rule, since individual
optimization leads to the social optimum anyway, as was shown earlier.
If we are not in the symmetric case, the equilibrium will be determined
by min{c1 /v2 , c2 /v1 }. In this case, strict liability does not result in the social
optimum.
The resolution is to use the negligence rule. Under this doctrine, the court
establishes a level of due care, x. In general, this could be different for different
¯
parties, but that generality is not necessary for this particular case. If the system
fails, there is no liability if the level of care/effort meets or exceeds the due care
standard. If the level of care/effort was less than the due care standard, then
the party who exerted inadequate care/effort must pay the other the costs of
system failure.
Although the traditional analysis of the negligence rule assumes the courts
determine the due care standard, an alternative model could involve the insur-
ance companies setting a due care standard. For example, insurance companies
could offer a contract specifying that the insured would be reimbursed for the
costs of an accident only if he or she had exercised an appropriate standard of
due care.
Let x∗ be the socially optimal effort level; i.e., the level that solves
max (v1 + v2 )P (x) − (c1 + c2 )x.
x
10
11. It therefor satisfies the first-order condition
(v1 + v2 )P (x∗ ) = c1 + c2 .
We need to show that if the due care standard is set at x = x∗ , then x1 = x2 = x
¯ ¯
is a Nash equilibrium.2
To prove this, assume that x2 = x. We must show that the optimal choice
¯
for agent 1 is x1 = x1 . Certainly we will never have x1 > x since choosing
¯ ¯
x1 larger than x has no impact on the probability of system failure and incurs
¯
positive cost. Will agent 1 ever want to choose x1 < x? Agent 1’s objective
¯
function is
v1 P (x1 ) + (1 − P (x1 ))v2 − c1 x1 .
Computing the derivative, and using the concavity of P (x), we find
(v1 + v2 )P (x1 ) − c1 > (v1 + v2 )P (x∗ ) − c1 = c2 .
Hence agent 1 will want to increase his level of effort when x1 < x1 . Summa-
¯
rizing:
Fact 9 In the case of weakest link, strict liability is not adequate in general to
achieve the socially optimal level of effort, and one must use a negligence rule
to induce the optimal effort.
Again, this is a standard result in liability law, which was first established
by Brown [1973]; see Proposition 2.2 in Shavell [1987], page 40. The argument
given here is easily modified to show that the negligence rule induces optimal
behavior in the sum-of-efforts case as well, or for that matter, for any other
form P (x1 , x2 ).
8 Sequential moves
8.1 Total effort
Let us now assume that the agents move sequentially, where the agent who
moves second can observe the choice of the agent who moves first. The following
discussion is based on Varian [1994].
We assume that agent 1 moves first. The utility of agent 1 as a function of
his effort is given by,
U1 (x1 ) = v1 P (x1 + f2 (x1 )) − c1 x1 .
which can be written as
U1 (x1 ) = v1 P (x1 + max{¯2 − x1 , 0}) − c1 x1 .
x
2 Of course, there will be many other Nash equilibria as well, due to the weakest-link
technology. The legal due-care standard has the advantage of serving as a focal point to
choose the most efficient such equilibrium.
11
12. expected
reliability
v1P( x 2)
v1P( x 1) −c1x 1
x2 x1 x1
Figure 4: Sequential contribution in total efforts case.
We can also write this as
v1 P (¯2 ) − c1 x1
x for x1 ≤ x2
¯
U1 (x1 ) =
v1 P (x1 ) − c1 x1 for x1 ≥ x2 .
¯
It is clear from Figure 4 that there are two possible optima: either the first
agent exerts zero effort and achieves payoff v1 P (¯2 ) or he contributes x1 and
x ¯
achieves utility v1 P (¯1 ) − c1 x1 .
x ¯
Case 1. The agent with the lowest value of vi /ci moves first. In this case the
optimal choice by the first player is to choose zero effort. This is true since
v1 P (¯2 ) > v1 P (¯1 ) > v1 P (¯1 ) − c1 x1 .
x x x ¯
Case 2. The agent with the highest value of vi /ci is the first contributor. In
this case, either contributor may free ride. If the agents have tastes that
are very similar, then the first contributor will free ride on the second’s
contribution. However, if the first mover likes the public good much more
than the second, then the first mover may prefer to contribute the entire
amount of the public good himself.
Referring to Figure 4 we see that there are two possible subgame perfect
equilibria: one is the Nash equilibrium, in which the agent who has the highest
benefit-cost ratio does everything. The other equilibrium is where the agent who
has the lowest benefit-cost contributes everything. This equilibrium cannot be
a Nash equilibrium since the threat to free ride by the agent who likes the public
good most is not credible in the simultaneous-move game.
Fact 10 The equilibrium in the sequential-move, the total-effort game always
involves the same or less reliability than the simultaneous-move game.
12
13. Note that it is always advantageous to move first since there are only two
possible outcomes and the first mover gets to pick the one he prefers.
Fact 11 If you want to ensure the highest level of security in the sequential-
move game, then you should make sure that the agent with the lower benefit-cost
ratio moves first.
8.2 Best-effort and weakest-link
The best-effort case is the same as the total-effort case. The weakest-link case
is a bit more interesting. Since each agent realizes that the other agent will, at
most, match his effort, there is no point in choosing a higher level of effort than
the agent who cares the least about reliability. On the other hand, there is no
need to settle for one of the inefficient Nash equilibria either.
Fact 12 The unique equilibrium in the sequential-move game will be the Nash
equilibrium in the simultaneous-move game that has the highest level of security,
namely min(¯1 , x2 ).
x ¯
Hirshleifer [1983] recognizes this and uses it as an argument for selecting the
Nash equilibrium with the highest amount of the public good as the “reasonable”
outcome.
9 Adversaries
Let us now briefly consider what happens if there is an adversary who is trying
to increase the probability of system failure. First we consider the case of just
two players, then we move to looking at what happens with a team on each side.
We let x be the effort of the defender, and y the effort of the attacker. Effort
costs the defender c and the attacker d. The defender gets utility v if the system
works, and the attacker gets utility w if the system fails. We suppose that the
probability of failure depends on “net effort,” x − y, and that there is a maximal
effort x and y for each player.
ˆ ˆ
The optimization problems for the attacker and defender can be written as
max vP (x − y) − cx (14)
x
max w[1 − P (x − y)] − dy. (15)
y
The first-order conditions are
vP (x − y) = c (16)
wP (x − y) = d. (17)
Let G(·) be the inverse function of P (x − y). By the second-order condition
this has to be locally decreasing, and we will assume it is globally decreasing.
13
14. y y
x
x
f(d/w)
f(c/v)
f(c/v)
f(d/w)
Figure 5: Reaction functions in adversarial case.
We can then apply the inverse function to write the two reaction functions:
x−y = G(c/v) (18)
x−y = G(d/w). (19)
Of course, these are only the reaction functions for interior optima. Adding
in the boundary conditions gives us:
x = min{max{G(c/v) + y, 0}, x}
ˆ (20)
y = min{max{G(d/w) − x, 0}, y}.
ˆ (21)
We plot these reaction functions in Figure 5. Note that there are two possible
equilibrium configurations. If c/v < d/w, we have x∗ = G(c/v) and y ∗ = 0,
while if c/v > d/w we have x∗ = x and y ∗ = x − G(d/w).
ˆ ˆ
Intuitively, if the cost-benefit ratio of the defender is smaller than that of the
attacker, the attacker gives up, and the defender does just enough to keep him
at bay. If the ratio is reversed, the defender has to go all out, and the attacker
pushes to keep him there.
10 Sum of efforts and weakest link
In the sum-of-efforts case the reaction functions are:
n m
xi − yi = G(cj /vj ) (22)
i=1 i=1
n m
xi − yi = G(dj /wj ). (23)
i=1 i=1
Here the party with the lowest cost/benefit ratio exerts effort, while everyone
else free rides. This becomes a “battle between the champions.”
14
15. In the weakest link case, the conditions for optimality are:
min{x1 , . . . , xn } − min{y1 , . . . , ym } = G(cj /vj ) (24)
min{x1 , . . . , xn } − min{y1 , . . . , ym } = G(dj /wj ). (25)
As opposed to a “battle of champions” we now have a “battle between the
slackers,” as the outcome is determined by the weakest player on each tam.
Note that when technology is total effort, large teams have an advantage,
whereas weakest link technology confers an advantage to small teams.
11 Future work
There are several avenues worth exploring:
• To what extent to these results extend to the more general framework of
Cornes [1993] and Sandler and Hartley [2001]. The possibility of Pareto
improving transfers is particularly interesting. Though Cornes [1993] ex-
amined this in the context of income transfers, knowledge transfers would
be particularly interesting in our context.
• One case where transfers are important are when agents can subsidize
other agents’ actions, as in Varian [1994]. The subgame perfect equilib-
rium of “announce subsidies then choose actions” is Pareto efficient in the
case we examine.
• One could look at capacity constraints on the part of the agents. For
example, each agent could put in only one unit of effort. Similarly, one
could look at increasing marginal cost of effort.
• Imperfect information adds additional phenomena. For example, Herma-
lin [1998] shows that in a model with uncertainty about payoffs, an agent
may choose to move first in order to demonstrate to the other agent that
a particular choice is worthwhile. Hence “leadership” plays a role of sig-
naling to the other agents.
• Arce and Sandler [2001] examine how results change when a contribution
game’s structure moves in the direction of best shot or weakest link. This
sort of partial comparative statics exercise could be of interest in our
context as well.
• One could examine situations where there were communication costs among
the cooperating agents, a la team theory. If, for example, there is imper-
fect information about what others are doing, it might lead to less free
riding.
15
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John Brown. Toward an economic theory of liability. Journal of Legal Studies,
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Richard Cornes. Dyke maintenance and other stories: Some neglected types of
public goods. Quarterly Journal of Economics, 108(1):259–271, 1993.
Benjamin Hermalin. Towards and economic theory of leadership: Leading by
example. American Economic Review, 88:1188–1206, 1998.
Jack Hirshleifer. From weakest-link to best-shot: the voluntary provision of
public goods. Public Choice, 41:371–86, 1983.
Mancur Olson and Richard Zeckhauser. An economic theory of alliances. Review
of Economics and Stastistics, 48(3):266–279, 1966.
Todd Sandler and Keith Hartley. Economics of alliances: The lessons for col-
lective action. Journal of Economic Literature, 34:869–896, 2001.
Stoeven Shavell. Economic Analysis of Accident Law. Harvard University Press,
Cambridge, MA, 1987.
Hal R. Varian. Sequential provision of public goods. Journal of Public Eco-
nomics, 53:165–186, 1994.
16