A Trap for Social Inclusion:
Prejudice, Oligarchy, and Rivalry
Leonardo Boncinelli
[email protected]
Simone D’Alessandro
[email protected]
Department of Economics and Management, University of Pisa
Co-Evolution of Behaviors and Institutions Working Group
Santa Fe Institute
January 12-15, 2014
Introduction
I
Examples abound where processes of social inclusion have
determined widespread benefits for involved communities.
I
I
I
The Marsh Farm regeneration policy in Luton =⇒ Social
Inclusion through Capacitation
Palmela (Portugal) =⇒ ’Citizen Participation and Local
Development’
Inclusive Cities Observatory
http://www.uclg-cisdp.org/en/observatory
I
Remark: those experiences were beneficial for the whole
community where they took place, but they where quite
demanding, at least in the launch phase.
I
Evidently, there exists some kind of societal trap preventing
societies from reaching more inclusive configurations.
Introduction
I
Examples abound where processes of social inclusion have
determined widespread benefits for involved communities.
I
I
I
The Marsh Farm regeneration policy in Luton =⇒ Social
Inclusion through Capacitation
Palmela (Portugal) =⇒ ’Citizen Participation and Local
Development’
Inclusive Cities Observatory
http://www.uclg-cisdp.org/en/observatory
I
Remark: those experiences were beneficial for the whole
community where they took place, but they where quite
demanding, at least in the launch phase.
I
Evidently, there exists some kind of societal trap preventing
societies from reaching more inclusive configurations.
Introduction
I
Examples abound where processes of social inclusion have
determined widespread benefits for involved communities.
I
I
I
The Marsh Farm regeneration policy in Luton =⇒ Social
Inclusion through Capacitation
Palmela (Portugal) =⇒ ’Citizen Participation and Local
Development’
Inclusive Cities Observatory
http://www.uclg-cisdp.org/en/observatory
I
Remark: those experiences were beneficial for the whole
community where they took place, but they where quite
demanding, at least in the launch phase.
I
Evidently, there exists some kind of societal trap preventing
societies from reaching more inclusive configurations.
Introduction
I
Examples abound where processes of social inclusion have
determined widespread benefits for involved communities.
I
I
I
The Marsh Farm regeneration policy in Luton =⇒ Social
Inclusion through Capacitation
Palmela (Portugal) =⇒ ’Citizen Participation and Local
Development’
Inclusive Cities Observatory
http://www.uclg-cisdp.org/en/observatory
I
Remark: those experiences were beneficial for the whole
community where they took place, but they where quite
demanding, at least in the launch phase.
I
Evidently, there exists some kind of societal trap preventing
societies from reaching more inclusive configurations.
Luton, England: The case of Marsh Farm
I
The Marsh Farm area is located on the very fringe of the London
Metropolitan Area,
I
Community Empowerment Strategy, began in the early 1990s, is
an ongoing community-based regeneration program.
I
The main objective is to enable the people to improve themselves
and their neighbourhood through the construction of a
community of self-help.
I
The process was started in an informal environment by a group
of inhabitants.
I
The initiators managed to involve a very broad part of the
inhabitants from different social and ethnic backgrounds.
I
The policy experienced an increasing level of institutionalisation,
rising interest of the local and national authorities in the success
of the local practices.
What We Do
Our aim is:
I
to attempt an explanation accounting for the highlighted
discrepancy between social welfare accruing from a more
inclusive society and the ability of the interested community to
reach such societal configuration.
We provide a model where:
i. a resident population is divided in two groups, one with included
agents and the other with excluded agents;
ii. both types of agents choose a level of cooperative effort, which
generates a basket of benefits (partially rival and partially
excludable);
iii. additionally included agents decide whether to share the rights of
inclusion with excluded agents.
What We Do
Our aim is:
I
to attempt an explanation accounting for the highlighted
discrepancy between social welfare accruing from a more
inclusive society and the ability of the interested community to
reach such societal configuration.
We provide a model where:
i. a resident population is divided in two groups, one with included
agents and the other with excluded agents;
ii. both types of agents choose a level of cooperative effort, which
generates a basket of benefits (partially rival and partially
excludable);
iii. additionally included agents decide whether to share the rights of
inclusion with excluded agents.
Flow of Benefits under Segregation
A possible matrix of benefits accessibility:
E
I
I
E
1
0
h
`
i
I
E
0
1
e
I
The digit 1 at the ij-th entry means that the effort exerted by an
i-type of agent is benefited by a j-type of agent; the digit 0 means
that the effort is not benefited.
I
Is it the right model of interactions?
I
No, included agents and excluded agents live as segregated
groups
I
A complementarity between cooperative effort and social
inclusion is likely to emerge, thus leading to multiple equilibria
Flow of Benefits under Segregation
A possible matrix of benefits accessibility:
E
I
I
E
1
0
h
`
i
I
E
0
1
e
I
The digit 1 at the ij-th entry means that the effort exerted by an
i-type of agent is benefited by a j-type of agent; the digit 0 means
that the effort is not benefited.
I
Is it the right model of interactions?
I
No, included agents and excluded agents live as segregated
groups
I
A complementarity between cooperative effort and social
inclusion is likely to emerge, thus leading to multiple equilibria
Flow of Benefits under Segregation
A possible matrix of benefits accessibility:
E
I
I
E
1
0
h
`
i
I
E
0
1
e
I
The digit 1 at the ij-th entry means that the effort exerted by an
i-type of agent is benefited by a j-type of agent; the digit 0 means
that the effort is not benefited.
I
Is it the right model of interactions?
I
No, included agents and excluded agents live as segregated
groups
I
A complementarity between cooperative effort and social
inclusion is likely to emerge, thus leading to multiple equilibria
Flow of Benefits under Segregation
A possible matrix of benefits accessibility:
E
I
I
E
1
0
h
`
i
I
E
0
1
e
I
The digit 1 at the ij-th entry means that the effort exerted by an
i-type of agent is benefited by a j-type of agent; the digit 0 means
that the effort is not benefited.
I
Is it the right model of interactions?
I
No, included agents and excluded agents live as segregated
groups
I
A complementarity between cooperative effort and social
inclusion is likely to emerge, thus leading to multiple equilibria
Flow of Benefits under Segregation
A possible matrix of benefits accessibility:
E
I
I
E
1
0
h
`
i
I
E
0
1
e
I
The digit 1 at the ij-th entry means that the effort exerted by an
i-type of agent is benefited by a j-type of agent; the digit 0 means
that the effort is not benefited.
I
Is it the right model of interactions?
I
No, included agents and excluded agents live as segregated
groups
I
A complementarity between cooperative effort and social
inclusion is likely to emerge, thus leading to multiple equilibria
Flow of Benefits under Coexistence
A better matrix of benefits accessibility:
I
E
I
1
0
E
1
1
I
Here included agents also enjoy the benefits coming from effort
exerted by excluded agents, since they all live in the same area
(e.g., a city).
I
On the converse, excluded agents are kept out of enjoying some
benefits, whose access is limited to included agents only.
I
From strategic point of view, in this setup social exclusion turns
out to be a strictly dominant choice for included agents.
Flow of Benefits under Coexistence
A better matrix of benefits accessibility:
I
E
I
1
0
E
1
1
I
Here included agents also enjoy the benefits coming from effort
exerted by excluded agents, since they all live in the same area
(e.g., a city).
I
On the converse, excluded agents are kept out of enjoying some
benefits, whose access is limited to included agents only.
I
From strategic point of view, in this setup social exclusion turns
out to be a strictly dominant choice for included agents.
Flow of Benefits under Coexistence
A better matrix of benefits accessibility:
I
E
I
1
0
E
1
1
I
Here included agents also enjoy the benefits coming from effort
exerted by excluded agents, since they all live in the same area
(e.g., a city).
I
On the converse, excluded agents are kept out of enjoying some
benefits, whose access is limited to included agents only.
I
From strategic point of view, in this setup social exclusion turns
out to be a strictly dominant choice for included agents.
Best Reply under Coexistence
E
h
`
i
I
e
I
i and e refer to the choice to include or to exclude, while h and `
refer to high and low effort exerted by an excluded agent. Blue
square are best reply of included agents, while blue circles are
best replies for excluded agents. Yellow denotes non-best reply
choices.
I
The unique equilibrium in the above game is (e, `), where
excluded agents exert low effort and are kept excluded. How can
a public authority intervene to promote social inclusion in this
framework?
Best Reply under Coexistence
E
h
`
i
I
e
I
i and e refer to the choice to include or to exclude, while h and `
refer to high and low effort exerted by an excluded agent. Blue
square are best reply of included agents, while blue circles are
best replies for excluded agents. Yellow denotes non-best reply
choices.
I
The unique equilibrium in the above game is (e, `), where
excluded agents exert low effort and are kept excluded. How can
a public authority intervene to promote social inclusion in this
framework?
Obstacles
A first obstacle preventing societies from reaching social inclusion is
due to the inability of included agents to forecast the adjustment in
behavior that excluded agents will perform once included.
I
Expectations on future behavior of excluded agents are based on
past behavior.
I
There is a prejudice which relates behavior to people, and not to
conditions in which people live.
I
a public authority may intervene in order to remove this
prejudice through participation programs (e.g. public
meetings, community planning) which make included agents
realize that the effort level of excluded agents will increase as a
consequence of their inclusion.
I
In the language of game theory, this amounts to force a change in
the game structure so to have a sequential choice of moves.
Obstacles
A first obstacle preventing societies from reaching social inclusion is
due to the inability of included agents to forecast the adjustment in
behavior that excluded agents will perform once included.
I
Expectations on future behavior of excluded agents are based on
past behavior.
I
There is a prejudice which relates behavior to people, and not to
conditions in which people live.
I
a public authority may intervene in order to remove this
prejudice through participation programs (e.g. public
meetings, community planning) which make included agents
realize that the effort level of excluded agents will increase as a
consequence of their inclusion.
I
In the language of game theory, this amounts to force a change in
the game structure so to have a sequential choice of moves.
Obstacles
A first obstacle preventing societies from reaching social inclusion is
due to the inability of included agents to forecast the adjustment in
behavior that excluded agents will perform once included.
I
Expectations on future behavior of excluded agents are based on
past behavior.
I
There is a prejudice which relates behavior to people, and not to
conditions in which people live.
I
a public authority may intervene in order to remove this
prejudice through participation programs (e.g. public
meetings, community planning) which make included agents
realize that the effort level of excluded agents will increase as a
consequence of their inclusion.
I
In the language of game theory, this amounts to force a change in
the game structure so to have a sequential choice of moves.
Obstacles
A first obstacle preventing societies from reaching social inclusion is
due to the inability of included agents to forecast the adjustment in
behavior that excluded agents will perform once included.
I
Expectations on future behavior of excluded agents are based on
past behavior.
I
There is a prejudice which relates behavior to people, and not to
conditions in which people live.
I
a public authority may intervene in order to remove this
prejudice through participation programs (e.g. public
meetings, community planning) which make included agents
realize that the effort level of excluded agents will increase as a
consequence of their inclusion.
I
In the language of game theory, this amounts to force a change in
the game structure so to have a sequential choice of moves.
Sequential game
I
i
e
E
E
h
`
h
`
I
Included agents realize that their choice to include or not will
affect the optimal level of effort chosen by excluded agent.
I
Even when the structure of moves is sequential, we are not sure
that social inclusion is beneficial for included agents.
Assumptions
I
nR number of residents, nI number of included agents, nK
number of excuded agents, nR = nI + nK
I
each resident chooses a level of effort e ∈ R+
One unit of effort yields a complex basket of benefits:
I
I
I
R
β αR + 1−α
nR
I
(1 − β) αI + 1−α
nI
where
I
I
I
I
I
β: measure of the share of benefits accruing to all residents
(non-excludable)
(1 − β): measure of the share of benefits accruing to included
only (excludable)
αR : degree of non-rivalry of the benefits accruing to all residents
αI : degree of non-rivalry of the benefits accruing to included only
Effort is costly, as described by a strictly convex and twice
differentiable cost function c(e), with c0 (e) > 0 and c00 (e) > 0.
Cost-benefit analysis
To understand the point, and the sources of difficulties, we should
focus on cost-benefit analysis for included agents.
I
An agent will find it convenient to exert low effort when
excluded, and high effort when included.
Therefore, for an already included agent the inclusion of an additional
person generates:
I
Benefit =⇒ The increase in the effort of newcomer.
I
Cost 1 =⇒ The overall benefits enjoyed by the group of
included agents must be shared with one person more.
I
Cost 2 =⇒ The optimal effort exerted by all included agents will
be slightly adjusted downwards, since now every included agent
will internalize a lower fraction of her effort.
Cost-benefit analysis
To understand the point, and the sources of difficulties, we should
focus on cost-benefit analysis for included agents.
I
An agent will find it convenient to exert low effort when
excluded, and high effort when included.
Therefore, for an already included agent the inclusion of an additional
person generates:
I
Benefit =⇒ The increase in the effort of newcomer.
I
Cost 1 =⇒ The overall benefits enjoyed by the group of
included agents must be shared with one person more.
I
Cost 2 =⇒ The optimal effort exerted by all included agents will
be slightly adjusted downwards, since now every included agent
will internalize a lower fraction of her effort.
Cost-benefit analysis
To understand the point, and the sources of difficulties, we should
focus on cost-benefit analysis for included agents.
I
An agent will find it convenient to exert low effort when
excluded, and high effort when included.
Therefore, for an already included agent the inclusion of an additional
person generates:
I
Benefit =⇒ The increase in the effort of newcomer.
I
Cost 1 =⇒ The overall benefits enjoyed by the group of
included agents must be shared with one person more.
I
Cost 2 =⇒ The optimal effort exerted by all included agents will
be slightly adjusted downwards, since now every included agent
will internalize a lower fraction of her effort.
Cost-benefit analysis
To understand the point, and the sources of difficulties, we should
focus on cost-benefit analysis for included agents.
I
An agent will find it convenient to exert low effort when
excluded, and high effort when included.
Therefore, for an already included agent the inclusion of an additional
person generates:
I
Benefit =⇒ The increase in the effort of newcomer.
I
Cost 1 =⇒ The overall benefits enjoyed by the group of
included agents must be shared with one person more.
I
Cost 2 =⇒ The optimal effort exerted by all included agents will
be slightly adjusted downwards, since now every included agent
will internalize a lower fraction of her effort.
Intuition
I
We observe that, the larger the group of included agents, the
lower the cost due to sharing with an additional agent.
I
Societies that are initially more oligarchic will find it more
problematic to undertake a social inclusion process.
I
The cost of extending the rights on inclusion can be thought of as
originating by the rivalry of benefits.
I
Goods and services that inclusion allows access to are only to
some extent rival.
I
If the degree of rivalry is reduced the cost of including an
additional agent decreases.
I
The degree of rivalry in the benefits reserved to the group of
included agents works as an obstacle to the rise of social
inclusion.
Intuition
I
We observe that, the larger the group of included agents, the
lower the cost due to sharing with an additional agent.
I
Societies that are initially more oligarchic will find it more
problematic to undertake a social inclusion process.
I
The cost of extending the rights on inclusion can be thought of as
originating by the rivalry of benefits.
I
Goods and services that inclusion allows access to are only to
some extent rival.
I
If the degree of rivalry is reduced the cost of including an
additional agent decreases.
I
The degree of rivalry in the benefits reserved to the group of
included agents works as an obstacle to the rise of social
inclusion.
Intuition
I
We observe that, the larger the group of included agents, the
lower the cost due to sharing with an additional agent.
I
Societies that are initially more oligarchic will find it more
problematic to undertake a social inclusion process.
I
The cost of extending the rights on inclusion can be thought of as
originating by the rivalry of benefits.
I
Goods and services that inclusion allows access to are only to
some extent rival.
I
If the degree of rivalry is reduced the cost of including an
additional agent decreases.
I
The degree of rivalry in the benefits reserved to the group of
included agents works as an obstacle to the rise of social
inclusion.
Intuition
I
We observe that, the larger the group of included agents, the
lower the cost due to sharing with an additional agent.
I
Societies that are initially more oligarchic will find it more
problematic to undertake a social inclusion process.
I
The cost of extending the rights on inclusion can be thought of as
originating by the rivalry of benefits.
I
Goods and services that inclusion allows access to are only to
some extent rival.
I
If the degree of rivalry is reduced the cost of including an
additional agent decreases.
I
The degree of rivalry in the benefits reserved to the group of
included agents works as an obstacle to the rise of social
inclusion.
Intuition
I
We observe that, the larger the group of included agents, the
lower the cost due to sharing with an additional agent.
I
Societies that are initially more oligarchic will find it more
problematic to undertake a social inclusion process.
I
The cost of extending the rights on inclusion can be thought of as
originating by the rivalry of benefits.
I
Goods and services that inclusion allows access to are only to
some extent rival.
I
If the degree of rivalry is reduced the cost of including an
additional agent decreases.
I
The degree of rivalry in the benefits reserved to the group of
included agents works as an obstacle to the rise of social
inclusion.
Results
Lemma
Any increase in social inclusion yields an increase in social welfare.
Thus, the optimal level of inclusion is nI = nR .
P ROPOSITION
Prejudice =⇒ If agents are naif, there is no room for the expansion of
social inclusion.
P ROPOSITION
Oligarchy =⇒ If αI > 0, there is a threshold level in the number of
socially included agents (n̂I ) beyond which an increase in social
inclusion is beneficial for socially included agents.
P ROPOSITION
Rivalry =⇒ An increase in the non-rival component of the benefit
accruing to socially included agents (αI ) reduces the threshold level
n̂I .
Results
Lemma
Any increase in social inclusion yields an increase in social welfare.
Thus, the optimal level of inclusion is nI = nR .
P ROPOSITION
Prejudice =⇒ If agents are naif, there is no room for the expansion of
social inclusion.
P ROPOSITION
Oligarchy =⇒ If αI > 0, there is a threshold level in the number of
socially included agents (n̂I ) beyond which an increase in social
inclusion is beneficial for socially included agents.
P ROPOSITION
Rivalry =⇒ An increase in the non-rival component of the benefit
accruing to socially included agents (αI ) reduces the threshold level
n̂I .
Results
Lemma
Any increase in social inclusion yields an increase in social welfare.
Thus, the optimal level of inclusion is nI = nR .
P ROPOSITION
Prejudice =⇒ If agents are naif, there is no room for the expansion of
social inclusion.
P ROPOSITION
Oligarchy =⇒ If αI > 0, there is a threshold level in the number of
socially included agents (n̂I ) beyond which an increase in social
inclusion is beneficial for socially included agents.
P ROPOSITION
Rivalry =⇒ An increase in the non-rival component of the benefit
accruing to socially included agents (αI ) reduces the threshold level
n̂I .
Results
Lemma
Any increase in social inclusion yields an increase in social welfare.
Thus, the optimal level of inclusion is nI = nR .
P ROPOSITION
Prejudice =⇒ If agents are naif, there is no room for the expansion of
social inclusion.
P ROPOSITION
Oligarchy =⇒ If αI > 0, there is a threshold level in the number of
socially included agents (n̂I ) beyond which an increase in social
inclusion is beneficial for socially included agents.
P ROPOSITION
Rivalry =⇒ An increase in the non-rival component of the benefit
accruing to socially included agents (αI ) reduces the threshold level
n̂I .
Results
ui (e∗ )
6
n̂I
nI
Policy Measures
I
Measure 1. Correcting prejudice =⇒ Participation Turn,
e.g. public meetings, community planning, converting a
simultaneous game to a sequential one where socially included
understand that the best reply of a new included agent is to
increase her effort.
I
Measure 2. Correcting oligarchy =⇒ to force the inclusion of
some residents, in order to let socially included agents to reach
threshold n̂I .
I
Measure 3. Correcting rivalry =⇒ to reduce the threshold,
increasing the non-rival component of benefits accruing to
socially included agents.
Policy Measures
I
Measure 1. Correcting prejudice =⇒ Participation Turn,
e.g. public meetings, community planning, converting a
simultaneous game to a sequential one where socially included
understand that the best reply of a new included agent is to
increase her effort.
I
Measure 2. Correcting oligarchy =⇒ to force the inclusion of
some residents, in order to let socially included agents to reach
threshold n̂I .
I
Measure 3. Correcting rivalry =⇒ to reduce the threshold,
increasing the non-rival component of benefits accruing to
socially included agents.
Policy Measures
I
Measure 1. Correcting prejudice =⇒ Participation Turn,
e.g. public meetings, community planning, converting a
simultaneous game to a sequential one where socially included
understand that the best reply of a new included agent is to
increase her effort.
I
Measure 2. Correcting oligarchy =⇒ to force the inclusion of
some residents, in order to let socially included agents to reach
threshold n̂I .
I
Measure 3. Correcting rivalry =⇒ to reduce the threshold,
increasing the non-rival component of benefits accruing to
socially included agents.
Future Steps
I
Question 1. Fairness =⇒ By increasing αI the difference in
utility between socially included and excluded increases. Is it a
right policy?
I
Question 2. Non-excludability =⇒ Which is the effect of an
increase in β?
I
Question 3. Rivarly and Excludability =⇒ can public authority
works modifying both αI and β in order to foster both social
inclusion and fairness.
Future Steps
I
Question 1. Fairness =⇒ By increasing αI the difference in
utility between socially included and excluded increases. Is it a
right policy?
I
Question 2. Non-excludability =⇒ Which is the effect of an
increase in β?
I
Question 3. Rivarly and Excludability =⇒ can public authority
works modifying both αI and β in order to foster both social
inclusion and fairness.
Future Steps
I
Question 1. Fairness =⇒ By increasing αI the difference in
utility between socially included and excluded increases. Is it a
right policy?
I
Question 2. Non-excludability =⇒ Which is the effect of an
increase in β?
I
Question 3. Rivarly and Excludability =⇒ can public authority
works modifying both αI and β in order to foster both social
inclusion and fairness.
Threshold towards social inclusion
red −→ low n̂I , yellow −→ high n̂I
black curves are level curves for n̂I
Fairness
difference in the utility levels between the two groups.
Fairness
difference in the utility levels between the two groups.
Conclusions
I
As stated in the objectives of Horizon 2020, building more
inclusive societies is crucial for further development of European
Union.
I
We shed some light on the discrepancy between:
I
I
I
We found that the emergence of societal traps can be explained
by three different sources:
I
I
I
I
social welfare accruing from a more inclusive society, and
the ability of the interested community to reach such societal
goal.
prejudice,
oligarchy, and
rivalry.
policy implications:
I
I
I
participation may help to eradicate prejudice,
a conflict between political goals may emerge,
fostering social inclusion may result in an increase in inequality.
Conclusions
I
As stated in the objectives of Horizon 2020, building more
inclusive societies is crucial for further development of European
Union.
I
We shed some light on the discrepancy between:
I
I
I
We found that the emergence of societal traps can be explained
by three different sources:
I
I
I
I
social welfare accruing from a more inclusive society, and
the ability of the interested community to reach such societal
goal.
prejudice,
oligarchy, and
rivalry.
policy implications:
I
I
I
participation may help to eradicate prejudice,
a conflict between political goals may emerge,
fostering social inclusion may result in an increase in inequality.
Conclusions
I
As stated in the objectives of Horizon 2020, building more
inclusive societies is crucial for further development of European
Union.
I
We shed some light on the discrepancy between:
I
I
I
We found that the emergence of societal traps can be explained
by three different sources:
I
I
I
I
social welfare accruing from a more inclusive society, and
the ability of the interested community to reach such societal
goal.
prejudice,
oligarchy, and
rivalry.
policy implications:
I
I
I
participation may help to eradicate prejudice,
a conflict between political goals may emerge,
fostering social inclusion may result in an increase in inequality.
Conclusions
I
As stated in the objectives of Horizon 2020, building more
inclusive societies is crucial for further development of European
Union.
I
We shed some light on the discrepancy between:
I
I
I
We found that the emergence of societal traps can be explained
by three different sources:
I
I
I
I
social welfare accruing from a more inclusive society, and
the ability of the interested community to reach such societal
goal.
prejudice,
oligarchy, and
rivalry.
policy implications:
I
I
I
participation may help to eradicate prejudice,
a conflict between political goals may emerge,
fostering social inclusion may result in an increase in inequality.
Utility
Utility of an excluded agent:
uk (e) =
X
j∈I
1 − αR
ej β αR +
nR
X
+
`∈K
1 − αR
β αR +
nR
e`
−c(ek )
Utility of an included agent:
ui (e) =
X
j∈I
+
X
`∈K
1 − αR
ej β αR +
nR
1 − αR
β αR +
nR
e`
1 − αI
+ (1 − β) αI +
nI
1 − αI
+ (1 − β) αI +
nI
where e = (e1 , . . . , enR ) is the vector of agents’ effort
+
− c(ei )
Utility
Utility of an excluded agent:
uk (e) =
X
j∈I
1 − αR
ej β αR +
nR
X
+
`∈K
1 − αR
β αR +
nR
e`
−c(ek )
Utility of an included agent:
ui (e) =
X
j∈I
+
X
`∈K
1 − αR
ej β αR +
nR
1 − αR
β αR +
nR
e`
1 − αI
+ (1 − β) αI +
nI
1 − αI
+ (1 − β) αI +
nI
where e = (e1 , . . . , enR ) is the vector of agents’ effort
+
− c(ei )
Assumptions
I
nR number of residents, nI number of included agents, nK
number of excuded agents, nR = nI + nK
I
each resident chooses a level of effort e ∈ R+
One unit of effort yields a complex basket of benefits:
I
I
I
R
β αR + 1−α
nR
I
(1 − β) αI + 1−α
nI
where
I
I
I
I
I
β: measure of the share of benefits accruing to all residents
(non-excludable)
(1 − β): measure of the share of benefits accruing to included
only (excludable)
αR : degree of non-rivalry of the benefits accruing to all residents
αI : degree of non-rivalry of the benefits accruing to included only
Effort is costly, as described by a strictly convex and twice
differentiable cost function c(e), with c0 (e) > 0 and c00 (e) > 0.
Assumptions
I
nR number of residents, nI number of included agents, nK
number of excuded agents, nR = nI + nK
I
each resident chooses a level of effort e ∈ R+
One unit of effort yields a complex basket of benefits:
I
I
I
R
β αR + 1−α
nR
I
(1 − β) αI + 1−α
nI
where
I
I
I
I
I
β: measure of the share of benefits accruing to all residents
(non-excludable)
(1 − β): measure of the share of benefits accruing to included
only (excludable)
αR : degree of non-rivalry of the benefits accruing to all residents
αI : degree of non-rivalry of the benefits accruing to included only
Effort is costly, as described by a strictly convex and twice
differentiable cost function c(e), with c0 (e) > 0 and c00 (e) > 0.
Assumptions
I
nR number of residents, nI number of included agents, nK
number of excuded agents, nR = nI + nK
I
each resident chooses a level of effort e ∈ R+
One unit of effort yields a complex basket of benefits:
I
I
I
R
β αR + 1−α
nR
I
(1 − β) αI + 1−α
nI
where
I
I
I
I
I
β: measure of the share of benefits accruing to all residents
(non-excludable)
(1 − β): measure of the share of benefits accruing to included
only (excludable)
αR : degree of non-rivalry of the benefits accruing to all residents
αI : degree of non-rivalry of the benefits accruing to included only
Effort is costly, as described by a strictly convex and twice
differentiable cost function c(e), with c0 (e) > 0 and c00 (e) > 0.
Assumptions
I
nR number of residents, nI number of included agents, nK
number of excuded agents, nR = nI + nK
I
each resident chooses a level of effort e ∈ R+
One unit of effort yields a complex basket of benefits:
I
I
I
R
β αR + 1−α
nR
I
(1 − β) αI + 1−α
nI
where
I
I
I
I
I
β: measure of the share of benefits accruing to all residents
(non-excludable)
(1 − β): measure of the share of benefits accruing to included
only (excludable)
αR : degree of non-rivalry of the benefits accruing to all residents
αI : degree of non-rivalry of the benefits accruing to included only
Effort is costly, as described by a strictly convex and twice
differentiable cost function c(e), with c0 (e) > 0 and c00 (e) > 0.
Optimal Choice
Bi (ei )
6
#
#
c(e)
#
#
#
Bk (ek )
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
e∗k
e∗i
e
Effects of Inclusion
The inclusion of a socially excluded agent affects the utility of
socially included agents through three channels:
(+) the effort of the new included agent increases
(-) the rival component of excludable benefits is shared with an
additional agent
(-) the optimal effort of every socially included agent decreases
Effects of Inclusion
The inclusion of a socially excluded agent affects the utility of
socially included agents through three channels:
(+) the effort of the new included agent increases
(-) the rival component of excludable benefits is shared with an
additional agent
(-) the optimal effort of every socially included agent decreases
Effects of Inclusion
The inclusion of a socially excluded agent affects the utility of
socially included agents through three channels:
(+) the effort of the new included agent increases
(-) the rival component of excludable benefits is shared with an
additional agent
(-) the optimal effort of every socially included agent decreases
Effects of Inclusion
The inclusion of a socially excluded agent affects the utility of
socially included agents through three channels:
(+) the effort of the new included agent increases
(-) the rival component of excludable benefits is shared with an
additional agent
(-) the optimal effort of every socially included agent decreases
Utility
Utility of an included agent in equilibrium:
1 − αR
ui (e ) = β αR +
nR
− c(e∗I ),
∗
1 − αI
+ (1 − β) αI +
nI
Utility of an excluded agent in equilibrium:
1 − αR
uk (e ) = β αR +
(nI e∗I + nK e∗K ) − c(e∗K )
nR
∗
(nI e∗I + nK e∗K )
Utility
Utility of an included agent in equilibrium:
1 − αR
ui (e ) = β αR +
nR
− c(e∗I ),
∗
1 − αI
+ (1 − β) αI +
nI
Utility of an excluded agent in equilibrium:
1 − αR
uk (e ) = β αR +
(nI e∗I + nK e∗K ) − c(e∗K )
nR
∗
(nI e∗I + nK e∗K )