a trap for social inclusion: prejudice, oligarchy, and...

A Trap for Social Inclusion:Prejudice, Oligarchy, and Rivalry

Leonardo Boncinellil.boncinelli@ec.unipi.it

Simone D’Alessandros.dale@ec.unipi.it

Department of Economics and Management, University of Pisa

Co-Evolution of Behaviors and Institutions Working GroupSanta Fe Institute

January 12-15, 2014

Introduction

I Examples abound where processes of social inclusion havedetermined widespread benefits for involved communities.

I The Marsh Farm regeneration policy in Luton =⇒ SocialInclusion through Capacitation

I Palmela (Portugal) =⇒ ’Citizen Participation and LocalDevelopment’

I Inclusive Cities Observatoryhttp://www.uclg-cisdp.org/en/observatory

I Remark: those experiences were beneficial for the wholecommunity where they took place, but they where quitedemanding, at least in the launch phase.

I Evidently, there exists some kind of societal trap preventingsocieties from reaching more inclusive configurations.

Introduction

I Examples abound where processes of social inclusion havedetermined widespread benefits for involved communities.

I The Marsh Farm regeneration policy in Luton =⇒ SocialInclusion through Capacitation

I Palmela (Portugal) =⇒ ’Citizen Participation and LocalDevelopment’

I Inclusive Cities Observatoryhttp://www.uclg-cisdp.org/en/observatory

I Remark: those experiences were beneficial for the wholecommunity where they took place, but they where quitedemanding, at least in the launch phase.

I Evidently, there exists some kind of societal trap preventingsocieties from reaching more inclusive configurations.

Introduction

I Examples abound where processes of social inclusion havedetermined widespread benefits for involved communities.

I The Marsh Farm regeneration policy in Luton =⇒ SocialInclusion through Capacitation

I Palmela (Portugal) =⇒ ’Citizen Participation and LocalDevelopment’

I Inclusive Cities Observatoryhttp://www.uclg-cisdp.org/en/observatory

I Remark: those experiences were beneficial for the wholecommunity where they took place, but they where quitedemanding, at least in the launch phase.

I Evidently, there exists some kind of societal trap preventingsocieties from reaching more inclusive configurations.

Introduction

I Examples abound where processes of social inclusion havedetermined widespread benefits for involved communities.

I The Marsh Farm regeneration policy in Luton =⇒ SocialInclusion through Capacitation

I Palmela (Portugal) =⇒ ’Citizen Participation and LocalDevelopment’

I Inclusive Cities Observatoryhttp://www.uclg-cisdp.org/en/observatory

I Remark: those experiences were beneficial for the wholecommunity where they took place, but they where quitedemanding, at least in the launch phase.

I Evidently, there exists some kind of societal trap preventingsocieties 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 LondonMetropolitan Area,

I Community Empowerment Strategy, began in the early 1990s, isan ongoing community-based regeneration program.

I The main objective is to enable the people to improve themselvesand their neighbourhood through the construction of acommunity of self-help.

I The process was started in an informal environment by a groupof inhabitants.

I The initiators managed to involve a very broad part of theinhabitants 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 successof 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 moreinclusive society and the ability of the interested community toreach such societal configuration.

We provide a model where:

i. a resident population is divided in two groups, one with includedagents and the other with excluded agents;

ii. both types of agents choose a level of cooperative effort, whichgenerates a basket of benefits (partially rival and partiallyexcludable);

iii. additionally included agents decide whether to share the rights ofinclusion 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 moreinclusive society and the ability of the interested community toreach such societal configuration.

We provide a model where:

i. a resident population is divided in two groups, one with includedagents and the other with excluded agents;

ii. both types of agents choose a level of cooperative effort, whichgenerates a basket of benefits (partially rival and partiallyexcludable);

iii. additionally included agents decide whether to share the rights ofinclusion with excluded agents.

Flow of Benefits under SegregationA possible matrix of benefits accessibility:

E

I

I E

0

1

1

0

e

iI

h `E

I The digit 1 at the ij-th entry means that the effort exerted by ani-type of agent is benefited by a j-type of agent; the digit 0 meansthat the effort is not benefited.

I Is it the right model of interactions?I No, included agents and excluded agents live as segregated

groupsI A complementarity between cooperative effort and social

inclusion is likely to emerge, thus leading to multiple equilibria

Flow of Benefits under SegregationA possible matrix of benefits accessibility:

E

I

I E

0

1

1

0

e

iI

h `E

I The digit 1 at the ij-th entry means that the effort exerted by ani-type of agent is benefited by a j-type of agent; the digit 0 meansthat the effort is not benefited.

I Is it the right model of interactions?I No, included agents and excluded agents live as segregated

groupsI A complementarity between cooperative effort and social

inclusion is likely to emerge, thus leading to multiple equilibria

Flow of Benefits under SegregationA possible matrix of benefits accessibility:

E

I

I E

0

1

1

0

e

iI

h `E

I The digit 1 at the ij-th entry means that the effort exerted by ani-type of agent is benefited by a j-type of agent; the digit 0 meansthat the effort is not benefited.

I Is it the right model of interactions?I No, included agents and excluded agents live as segregated

groupsI A complementarity between cooperative effort and social

inclusion is likely to emerge, thus leading to multiple equilibria

Flow of Benefits under SegregationA possible matrix of benefits accessibility:

E

I

I E

0

1

1

0

e

iI

h `E

I The digit 1 at the ij-th entry means that the effort exerted by ani-type of agent is benefited by a j-type of agent; the digit 0 meansthat the effort is not benefited.

I Is it the right model of interactions?I No, included agents and excluded agents live as segregated

groupsI A complementarity between cooperative effort and social

inclusion is likely to emerge, thus leading to multiple equilibria

Flow of Benefits under SegregationA possible matrix of benefits accessibility:

E

I

I E

0

1

1

0

e

iI

h `E

I The digit 1 at the ij-th entry means that the effort exerted by ani-type of agent is benefited by a j-type of agent; the digit 0 meansthat the effort is not benefited.

I Is it the right model of interactions?I No, included agents and excluded agents live as segregated

groupsI 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:

E

I

I E

1

1

1

0

I Here included agents also enjoy the benefits coming from effortexerted 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 somebenefits, whose access is limited to included agents only.

I From strategic point of view, in this setup social exclusion turnsout to be a strictly dominant choice for included agents.

Flow of Benefits under Coexistence

A better matrix of benefits accessibility:

E

I

I E

1

1

1

0

I Here included agents also enjoy the benefits coming from effortexerted 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 somebenefits, whose access is limited to included agents only.

I From strategic point of view, in this setup social exclusion turnsout to be a strictly dominant choice for included agents.

Flow of Benefits under Coexistence

A better matrix of benefits accessibility:

E

I

I E

1

1

1

0

I Here included agents also enjoy the benefits coming from effortexerted 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 somebenefits, whose access is limited to included agents only.

I From strategic point of view, in this setup social exclusion turnsout to be a strictly dominant choice for included agents.

Best Reply under Coexistence

e

iI

h `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. Bluesquare are best reply of included agents, while blue circles arebest replies for excluded agents. Yellow denotes non-best replychoices.

I The unique equilibrium in the above game is (e, `), whereexcluded agents exert low effort and are kept excluded. How cana public authority intervene to promote social inclusion in thisframework?

Best Reply under Coexistence

e

iI

h `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. Bluesquare are best reply of included agents, while blue circles arebest replies for excluded agents. Yellow denotes non-best replychoices.

I The unique equilibrium in the above game is (e, `), whereexcluded agents exert low effort and are kept excluded. How cana public authority intervene to promote social inclusion in thisframework?

Obstacles

A first obstacle preventing societies from reaching social inclusion isdue to the inability of included agents to forecast the adjustment inbehavior that excluded agents will perform once included.

I Expectations on future behavior of excluded agents are based onpast behavior.

I There is a prejudice which relates behavior to people, and not toconditions in which people live.

I a public authority may intervene in order to remove thisprejudice through participation programs (e.g. publicmeetings, community planning) which make included agentsrealize that the effort level of excluded agents will increase as aconsequence of their inclusion.

I In the language of game theory, this amounts to force a change inthe game structure so to have a sequential choice of moves.

Obstacles

A first obstacle preventing societies from reaching social inclusion isdue to the inability of included agents to forecast the adjustment inbehavior that excluded agents will perform once included.

I Expectations on future behavior of excluded agents are based onpast behavior.

I There is a prejudice which relates behavior to people, and not toconditions in which people live.

I a public authority may intervene in order to remove thisprejudice through participation programs (e.g. publicmeetings, community planning) which make included agentsrealize that the effort level of excluded agents will increase as aconsequence of their inclusion.

I In the language of game theory, this amounts to force a change inthe game structure so to have a sequential choice of moves.

Obstacles

A first obstacle preventing societies from reaching social inclusion isdue to the inability of included agents to forecast the adjustment inbehavior that excluded agents will perform once included.

I Expectations on future behavior of excluded agents are based onpast behavior.

I There is a prejudice which relates behavior to people, and not toconditions in which people live.

I a public authority may intervene in order to remove thisprejudice through participation programs (e.g. publicmeetings, community planning) which make included agentsrealize that the effort level of excluded agents will increase as aconsequence of their inclusion.

I In the language of game theory, this amounts to force a change inthe game structure so to have a sequential choice of moves.

Obstacles

A first obstacle preventing societies from reaching social inclusion isdue to the inability of included agents to forecast the adjustment inbehavior that excluded agents will perform once included.

I Expectations on future behavior of excluded agents are based onpast behavior.

I There is a prejudice which relates behavior to people, and not toconditions in which people live.

I a public authority may intervene in order to remove thisprejudice through participation programs (e.g. publicmeetings, community planning) which make included agentsrealize that the effort level of excluded agents will increase as aconsequence of their inclusion.

I In the language of game theory, this amounts to force a change inthe game structure so to have a sequential choice of moves.

Sequential game

I

E

h `

i

E

h `

e

I Included agents realize that their choice to include or not willaffect the optimal level of effort chosen by excluded agent.

I Even when the structure of moves is sequential, we are not surethat 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+I One unit of effort yields a complex basket of benefits:

I β(αR + 1−αR

nR

)I (1− β)

(αI + 1−αI

nI

)where

I β: measure of the share of benefits accruing to all residents(non-excludable)

I (1− β): measure of the share of benefits accruing to includedonly (excludable)

I αR: degree of non-rivalry of the benefits accruing to all residentsI αI : degree of non-rivalry of the benefits accruing to included only

I Effort is costly, as described by a strictly convex and twicedifferentiable cost function c(e), with c′(e) > 0 and c′′(e) > 0.

Cost-benefit analysis

To understand the point, and the sources of difficulties, we shouldfocus on cost-benefit analysis for included agents.

I An agent will find it convenient to exert low effort whenexcluded, and high effort when included.

Therefore, for an already included agent the inclusion of an additionalperson 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 agentwill internalize a lower fraction of her effort.

Cost-benefit analysis

To understand the point, and the sources of difficulties, we shouldfocus on cost-benefit analysis for included agents.

I An agent will find it convenient to exert low effort whenexcluded, and high effort when included.

Therefore, for an already included agent the inclusion of an additionalperson 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 agentwill internalize a lower fraction of her effort.

Cost-benefit analysis

To understand the point, and the sources of difficulties, we shouldfocus on cost-benefit analysis for included agents.

I An agent will find it convenient to exert low effort whenexcluded, and high effort when included.

Therefore, for an already included agent the inclusion of an additionalperson 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 agentwill internalize a lower fraction of her effort.

Cost-benefit analysis

To understand the point, and the sources of difficulties, we shouldfocus on cost-benefit analysis for included agents.

I An agent will find it convenient to exert low effort whenexcluded, and high effort when included.

Therefore, for an already included agent the inclusion of an additionalperson 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 agentwill internalize a lower fraction of her effort.

Intuition

I We observe that, the larger the group of included agents, thelower the cost due to sharing with an additional agent.

I Societies that are initially more oligarchic will find it moreproblematic to undertake a social inclusion process.

I The cost of extending the rights on inclusion can be thought of asoriginating by the rivalry of benefits.

I Goods and services that inclusion allows access to are only tosome extent rival.

I If the degree of rivalry is reduced the cost of including anadditional agent decreases.

I The degree of rivalry in the benefits reserved to the group ofincluded agents works as an obstacle to the rise of socialinclusion.

Intuition

I We observe that, the larger the group of included agents, thelower the cost due to sharing with an additional agent.

I Societies that are initially more oligarchic will find it moreproblematic to undertake a social inclusion process.

I The cost of extending the rights on inclusion can be thought of asoriginating by the rivalry of benefits.

I Goods and services that inclusion allows access to are only tosome extent rival.

I If the degree of rivalry is reduced the cost of including anadditional agent decreases.

I The degree of rivalry in the benefits reserved to the group ofincluded agents works as an obstacle to the rise of socialinclusion.

Intuition

I We observe that, the larger the group of included agents, thelower the cost due to sharing with an additional agent.

I Societies that are initially more oligarchic will find it moreproblematic to undertake a social inclusion process.

I The cost of extending the rights on inclusion can be thought of asoriginating by the rivalry of benefits.

I Goods and services that inclusion allows access to are only tosome extent rival.

I If the degree of rivalry is reduced the cost of including anadditional agent decreases.

I The degree of rivalry in the benefits reserved to the group ofincluded agents works as an obstacle to the rise of socialinclusion.

Intuition

I We observe that, the larger the group of included agents, thelower the cost due to sharing with an additional agent.

I Societies that are initially more oligarchic will find it moreproblematic to undertake a social inclusion process.

I The cost of extending the rights on inclusion can be thought of asoriginating by the rivalry of benefits.

I Goods and services that inclusion allows access to are only tosome extent rival.

I If the degree of rivalry is reduced the cost of including anadditional agent decreases.

I The degree of rivalry in the benefits reserved to the group ofincluded agents works as an obstacle to the rise of socialinclusion.

Intuition

I We observe that, the larger the group of included agents, thelower the cost due to sharing with an additional agent.

I Societies that are initially more oligarchic will find it moreproblematic to undertake a social inclusion process.

I The cost of extending the rights on inclusion can be thought of asoriginating by the rivalry of benefits.

I Goods and services that inclusion allows access to are only tosome extent rival.

I If the degree of rivalry is reduced the cost of including anadditional agent decreases.

I The degree of rivalry in the benefits reserved to the group ofincluded agents works as an obstacle to the rise of socialinclusion.

Results

LemmaAny increase in social inclusion yields an increase in social welfare.Thus, the optimal level of inclusion is nI = nR.

PROPOSITIONPrejudice =⇒ If agents are naif, there is no room for the expansion ofsocial inclusion.

PROPOSITIONOligarchy =⇒ If αI > 0, there is a threshold level in the number ofsocially included agents (n̂I) beyond which an increase in socialinclusion is beneficial for socially included agents.

PROPOSITIONRivalry =⇒ An increase in the non-rival component of the benefitaccruing to socially included agents (αI) reduces the threshold leveln̂I .

Results

LemmaAny increase in social inclusion yields an increase in social welfare.Thus, the optimal level of inclusion is nI = nR.

PROPOSITIONPrejudice =⇒ If agents are naif, there is no room for the expansion ofsocial inclusion.

PROPOSITIONOligarchy =⇒ If αI > 0, there is a threshold level in the number ofsocially included agents (n̂I) beyond which an increase in socialinclusion is beneficial for socially included agents.

PROPOSITIONRivalry =⇒ An increase in the non-rival component of the benefitaccruing to socially included agents (αI) reduces the threshold leveln̂I .

Results

LemmaAny increase in social inclusion yields an increase in social welfare.Thus, the optimal level of inclusion is nI = nR.

PROPOSITIONPrejudice =⇒ If agents are naif, there is no room for the expansion ofsocial inclusion.

PROPOSITIONOligarchy =⇒ If αI > 0, there is a threshold level in the number ofsocially included agents (n̂I) beyond which an increase in socialinclusion is beneficial for socially included agents.

PROPOSITIONRivalry =⇒ An increase in the non-rival component of the benefitaccruing to socially included agents (αI) reduces the threshold leveln̂I .

Results

LemmaAny increase in social inclusion yields an increase in social welfare.Thus, the optimal level of inclusion is nI = nR.

PROPOSITIONPrejudice =⇒ If agents are naif, there is no room for the expansion ofsocial inclusion.

PROPOSITIONOligarchy =⇒ If αI > 0, there is a threshold level in the number ofsocially included agents (n̂I) beyond which an increase in socialinclusion is beneficial for socially included agents.

PROPOSITIONRivalry =⇒ An increase in the non-rival component of the benefitaccruing to socially included agents (αI) reduces the threshold leveln̂I .

Results

-

6

nI

ui(e∗)

n̂I

Policy Measures

I Measure 1. Correcting prejudice =⇒ Participation Turn,e.g. public meetings, community planning, converting asimultaneous game to a sequential one where socially includedunderstand that the best reply of a new included agent is toincrease her effort.

I Measure 2. Correcting oligarchy =⇒ to force the inclusion ofsome residents, in order to let socially included agents to reachthreshold n̂I .

I Measure 3. Correcting rivalry =⇒ to reduce the threshold,increasing the non-rival component of benefits accruing tosocially included agents.

Policy Measures

I Measure 1. Correcting prejudice =⇒ Participation Turn,e.g. public meetings, community planning, converting asimultaneous game to a sequential one where socially includedunderstand that the best reply of a new included agent is toincrease her effort.

I Measure 2. Correcting oligarchy =⇒ to force the inclusion ofsome residents, in order to let socially included agents to reachthreshold n̂I .

I Measure 3. Correcting rivalry =⇒ to reduce the threshold,increasing the non-rival component of benefits accruing tosocially included agents.

Policy Measures

I Measure 1. Correcting prejudice =⇒ Participation Turn,e.g. public meetings, community planning, converting asimultaneous game to a sequential one where socially includedunderstand that the best reply of a new included agent is toincrease her effort.

I Measure 2. Correcting oligarchy =⇒ to force the inclusion ofsome residents, in order to let socially included agents to reachthreshold n̂I .

I Measure 3. Correcting rivalry =⇒ to reduce the threshold,increasing the non-rival component of benefits accruing tosocially included agents.

Future Steps

I Question 1. Fairness =⇒ By increasing αI the difference inutility between socially included and excluded increases. Is it aright policy?

I Question 2. Non-excludability =⇒Which is the effect of anincrease in β?

I Question 3. Rivarly and Excludability =⇒ can public authorityworks modifying both αI and β in order to foster both socialinclusion and fairness.

Future Steps

I Question 1. Fairness =⇒ By increasing αI the difference inutility between socially included and excluded increases. Is it aright policy?

I Question 2. Non-excludability =⇒Which is the effect of anincrease in β?

I Question 3. Rivarly and Excludability =⇒ can public authorityworks modifying both αI and β in order to foster both socialinclusion and fairness.

Future Steps

I Question 1. Fairness =⇒ By increasing αI the difference inutility between socially included and excluded increases. Is it aright policy?

I Question 2. Non-excludability =⇒Which is the effect of anincrease in β?

I Question 3. Rivarly and Excludability =⇒ can public authorityworks modifying both αI and β in order to foster both socialinclusion 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 moreinclusive societies is crucial for further development of EuropeanUnion.

I We shed some light on the discrepancy between:I social welfare accruing from a more inclusive society, andI the ability of the interested community to reach such societal

goal.I We found that the emergence of societal traps can be explained

by three different sources:I prejudice,I oligarchy, andI rivalry.

I policy implications:I participation may help to eradicate prejudice,I a conflict between political goals may emerge,I fostering social inclusion may result in an increase in inequality.

Conclusions

I As stated in the objectives of Horizon 2020, building moreinclusive societies is crucial for further development of EuropeanUnion.

I We shed some light on the discrepancy between:I social welfare accruing from a more inclusive society, andI the ability of the interested community to reach such societal

goal.I We found that the emergence of societal traps can be explained

by three different sources:I prejudice,I oligarchy, andI rivalry.

I policy implications:I participation may help to eradicate prejudice,I a conflict between political goals may emerge,I fostering social inclusion may result in an increase in inequality.

Conclusions

I As stated in the objectives of Horizon 2020, building moreinclusive societies is crucial for further development of EuropeanUnion.

I We shed some light on the discrepancy between:I social welfare accruing from a more inclusive society, andI the ability of the interested community to reach such societal

goal.I We found that the emergence of societal traps can be explained

by three different sources:I prejudice,I oligarchy, andI rivalry.

I policy implications:I participation may help to eradicate prejudice,I a conflict between political goals may emerge,I fostering social inclusion may result in an increase in inequality.

Conclusions

I As stated in the objectives of Horizon 2020, building moreinclusive societies is crucial for further development of EuropeanUnion.

I We shed some light on the discrepancy between:I social welfare accruing from a more inclusive society, andI the ability of the interested community to reach such societal

goal.I We found that the emergence of societal traps can be explained

by three different sources:I prejudice,I oligarchy, andI rivalry.

I policy implications:I participation may help to eradicate prejudice,I a conflict between political goals may emerge,I fostering social inclusion may result in an increase in inequality.

Utility

Utility of an excluded agent:

uk(e) =∑j∈I

ej

[β

(αR + 1− αR

nR

)]+∑`∈K

e`

[β

(αR + 1− αR

nR

)]−c(ek)

Utility of an included agent:

ui(e) =∑j∈I

ej

[β

(αR + 1− αR

nR

)+ (1− β)

(αI + 1− αI

nI

)]+

+∑`∈K

e`

[β

(αR + 1− αR

nR

)+ (1− β)

(αI + 1− αI

nI

)]− c(ei)

where e = (e1, . . . , enR) is the vector of agents’ effort

Utility

Utility of an excluded agent:

uk(e) =∑j∈I

ej

[β

(αR + 1− αR

nR

)]+∑`∈K

e`

[β

(αR + 1− αR

nR

)]−c(ek)

Utility of an included agent:

ui(e) =∑j∈I

ej

[β

(αR + 1− αR

nR

)+ (1− β)

(αI + 1− αI

nI

)]+

+∑`∈K

e`

[β

(αR + 1− αR

nR

)+ (1− β)

(αI + 1− αI

nI

)]− c(ei)

where e = (e1, . . . , enR) is the vector of agents’ effort

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+I One unit of effort yields a complex basket of benefits:

I β(αR + 1−αR

nR

)I (1− β)

(αI + 1−αI

nI

)where

I β: measure of the share of benefits accruing to all residents(non-excludable)

I (1− β): measure of the share of benefits accruing to includedonly (excludable)

I αR: degree of non-rivalry of the benefits accruing to all residentsI αI : degree of non-rivalry of the benefits accruing to included only

I Effort is costly, as described by a strictly convex and twicedifferentiable cost function c(e), with c′(e) > 0 and c′′(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+I One unit of effort yields a complex basket of benefits:

I β(αR + 1−αR

nR

)I (1− β)

(αI + 1−αI

nI

)where

I β: measure of the share of benefits accruing to all residents(non-excludable)

I (1− β): measure of the share of benefits accruing to includedonly (excludable)

I αR: degree of non-rivalry of the benefits accruing to all residentsI αI : degree of non-rivalry of the benefits accruing to included only

I Effort is costly, as described by a strictly convex and twicedifferentiable cost function c(e), with c′(e) > 0 and c′′(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+I One unit of effort yields a complex basket of benefits:

I β(αR + 1−αR

nR

)I (1− β)

(αI + 1−αI

nI

)where

I β: measure of the share of benefits accruing to all residents(non-excludable)

I (1− β): measure of the share of benefits accruing to includedonly (excludable)

I αR: degree of non-rivalry of the benefits accruing to all residentsI αI : degree of non-rivalry of the benefits accruing to included only

I Effort is costly, as described by a strictly convex and twicedifferentiable cost function c(e), with c′(e) > 0 and c′′(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+I One unit of effort yields a complex basket of benefits:

I β(αR + 1−αR

nR

)I (1− β)

(αI + 1−αI

nI

)where

I β: measure of the share of benefits accruing to all residents(non-excludable)

I (1− β): measure of the share of benefits accruing to includedonly (excludable)

I αR: degree of non-rivalry of the benefits accruing to all residentsI αI : degree of non-rivalry of the benefits accruing to included only

I Effort is costly, as described by a strictly convex and twicedifferentiable cost function c(e), with c′(e) > 0 and c′′(e) > 0.

Optimal Choice

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6

Bk(ek)

Bi(ei)

ee∗k

c(e)

e∗i

Effects of Inclusion

The inclusion of a socially excluded agent affects the utility ofsocially included agents through three channels:

(+) the effort of the new included agent increases

(-) the rival component of excludable benefits is shared with anadditional agent

(-) the optimal effort of every socially included agent decreases

Effects of Inclusion

The inclusion of a socially excluded agent affects the utility ofsocially included agents through three channels:

(+) the effort of the new included agent increases

(-) the rival component of excludable benefits is shared with anadditional agent

(-) the optimal effort of every socially included agent decreases

Effects of Inclusion

The inclusion of a socially excluded agent affects the utility ofsocially included agents through three channels:

(+) the effort of the new included agent increases

(-) the rival component of excludable benefits is shared with anadditional agent

(-) the optimal effort of every socially included agent decreases

Effects of Inclusion

The inclusion of a socially excluded agent affects the utility ofsocially included agents through three channels:

(+) the effort of the new included agent increases

(-) the rival component of excludable benefits is shared with anadditional agent

(-) the optimal effort of every socially included agent decreases

Utility

Utility of an included agent in equilibrium:

ui(e∗) =[β

(αR + 1− αR

nR

)+ (1− β)

(αI + 1− αI

nI

)](nIe∗I + nKe∗K)

− c(e∗I ),

Utility of an excluded agent in equilibrium:

uk(e∗) = β

(αR + 1− αR

nR

)(nIe∗I + nKe∗K)− c(e∗K)

Utility

Utility of an included agent in equilibrium:

ui(e∗) =[β

(αR + 1− αR

nR

)+ (1− β)

(αI + 1− αI

nI

)](nIe∗I + nKe∗K)

− c(e∗I ),

Utility of an excluded agent in equilibrium:

uk(e∗) = β

(αR + 1− αR

nR

)(nIe∗I + nKe∗K)− c(e∗K)