Reasoning About Coalitional Effectivity in Modal Extension of Łukasiewicz Logic

Reasoning About Coalitional Effectivity in

Modal Extension of Łukasiewicz Logic

Tomáš Kroupa1 Bruno Teheux2

1_{Institute of Information Theory and Automation of the ASCR, Czech Republic}

Strategic game forms

G = hN, {Σ

| i ∈ N}, S, oi

N = {1, . . . , n} is a set ofplayers

Σi is thestrategy setof player i ∈ N

S is a set ofoutcome states

o :Q

Measuring the power of coalitions

G = hN, {Σi | i ∈ N}, S, oi

For acoalitionC ∈ PN : C stands for N \ C

σCstands for a tuple of strategies for players in C

Definition. Theeffectivity functionE[

G : PN → PPSassociated

with Gis defined by

Towards a quantitative notion of effectivity

G = hN, {Σi | i ∈ N}, S, oi

N = set of countries in EU

Σi =set of economic policies that country i can adopt

o(σ) = (ri)i∈N where ri =rate of employment of country i if

strategy vector σ is applied

The limitation of E[ G

An outcome sate (ri)i∈N is composed ofquantitativeinformation

while

E[

Towards a quantitative notion of effectivity

G = hN, {Σi | i ∈ N}, S, oi

N = set of countries in EU

Σi =set of economic policies that country i can adopt

o(σ) = (ri)i∈N where ri =rate of employment of country i if

strategy vector σ is applied

We define generalized effectivity functions

G = hN, {Σi | i ∈ N}, S, oi

Łk := {0,_k1, . . . ,k −1_k ,1} is the scale

Definition. TheŁk-valued effectivity function of G

Ł

-valued effectivity function in the EU example

the minimum rate of employment for coun-tries in C that coalition C can enforce

‘employment sovereignty of C’

meani∈N ri

the mean country level rate of employment in EU that coalition C can enforce

The poster session

1. Examples illustrating the relevance of Łk-valued effectivity

function

2. Axiomatization of functions E : PN × ŁS_k → Łk that are

playable, i.e., for which there is a G such that E = E_G

F Łk-valued version ofsuperadditivity F

3. A modal language to talk about Łk-valued effectivity

functions (based on Łukasiewicz logic)

4. A Łn-valued neighborhood semantic and acompleteness