MONOTONE PROPERTIES OF INFORMATION CONTROL IN A GAME WITH UNCERTAINTY Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, xii, 113-127

Monotone Properties of Information Control in a Game with

Uncertainty

Denis Fedyanin

IEEE member, V.A.Trapeznikov Institute of Control Sciences, Moscow, National Research University Higher School of Economics, Russian Federation, Moscow

E-mail: df edyanin@inbox. ru

Abstract This study sets out to investigate the impact of information control. We used our previous reflexive analysis of a game to find the sensitivity of strategies and utility functions to increasing beliefs about thresholds. The game itself is constructed by using a normal form game and making suggestions on the agents's believes and knowledge weaker. We found domains of parameters where monotonicity of the impact holds too. Together, these results provide important insights into the impact of reflexive analysis on the properties of information control.

1. Introduction

Let's say there are a set of agents N = {1,..., n}, a set real, non-negative strategies X = {X1;..., Xn}, and a set of utility functions F(A) = {/i(A),..., /n(A)} with a parameter A. One can consider a game

G =< N,X,F(A) > .

We have investigated a case when agents don't have consensus on the value of

A

y = ±1 p | -y | (y ^ | Kiy | Biy | Cky | Cby,

where ± is False. Elementary propositions are elements of a set P € {(A = x)|x € R}. {Kiy} is a set of knowledge operators of agents that describes their knowledge about a value y. {Biy} is a set of belief operators of agents that describes their beliefs about a value CK y means that y is common knowledge among agents in N. CB y means that y is a common belief among agents in N. We will write G/ =< N,X,F(A), I > for a game G =< N,X,F(A) > with a given logic assumptions or axioms of informational structure I. The ordinary case is G =< N, X, F(A), CK(A = Ao) >, where A0 is an actual value of a parameter A. It is just a game G =< N, X, F(A0) > in a normal form. Note that Nash equilibria for Gck =< N,X,F(A), Ck (A = Ao) > and Gcb =< N,X,F(A),Cb (A = Ao) > coincides though resulting values of utility functions could differ since CK (A = A0) ^ (A = A0) but there is no such theorem for CB (A = A0). A belief could be false even if it is a common belief.

There is a well-known way to investigate this game using Nash equilibria. Each Nash equilibrium is a vector y = (y1;..., yn) such that Vxi € Xi

/i(y 1, ...,yi-i,yi,yi+i, ...,yn) > /i(yi, ..., yi-1 ,xi, yi+1, ...,yn)

We denote G0 =< N,X,F(A),y > ^g. GViB-(A=A ) =< N,X,F(A), V«Bi(A = Ai) >

We knew the equilibria for some games (Fedyanin, 2019). There were functional dependencies strategies and utility on beliefs about the parameter A. So we found the intervals of monotonicity using derivatives of the functional dependencies. Some expressions were very obvious or easy to find but others were very difficult for analysis. We expressed our results in several theorems.

2. An example of game

We will continue investigations of a game of collective actions (Fedyanin and Chkhar-tishvili, 2011). There are a set of agents N = {1,..., n}, a set of real not negative strategies and a set of utility functions

where 0 < r < 1.

The corresponding practical interpretation lies in that the agents apply the strategies and it appears successful (provides a positive contribution to the utility-functions of the agents) when the total effort exceeds a specific threshold; the latter is set equal to 1. With the strategy being successful, the agent's gain(the first term in utility function) increases with the increasing effort of the agent. On the other hand, the agent's effort itself results in a negative contribution to the utility function (see the second term) which depends on the type rj. The larger the type of variable, the "easier" the agent applies the strategy (for instance, in a psychological sense, it could be explained by the agent's greater loyalty or liking for the joint action) (Fedyanin and Chkhartishvili, 2011).

The Cournot oligopoly model (Cournot, 1960) looks similar but it is not the same because of different utility functions

The corresponding practical interpretation of the Cournot oligopoly is the following: strategies are the amounts of sold products, utility functions are the amounts of products multiplied by a price that decreases when the total amount of sold products increases minus costs.

There are some important differences that make the game of collective actions look like a combination of the Cournot oligopoly and the game theoretical modification of Granovetter (Granovetter, 1978) and not just the Cournot oligopoly. The Breer Threshold model (Breer et al., 2017) is the one where utility functions are

and a set of strategies is restricted to binary values - strategy is equal either 0 or 1. Anyway we can apply all ideas below for the Cournot oligopoly as well but we haven't applied them yet.

In this paper we propose to consider A as an uncertain parameter for agents and they have to make some suggestion about it.

3. Results

3.1. Players with common knowledge

We can model it by a game

Gck =< N, X, F(A),Ck (A = Ao) >

There is a well-known way to find the Nash equilibrium. It is to compose and solve a system of equations where the strategy of each player equals his or her best response

xi = BRi(x — i) = 2ri Vij - A | + a, Vi e N 1 — r i \ /

\j=i /

o, (£.=. x, - A > o

e¿ = '

where

_-T-ïî [I2j=i xj - A , T-H xj - A < 0

Zero Nash Equilibrium for GCk

Theorem 1. An existance of a zero Nash equilibrium doesn't depend on the value of the threshold in the game GCk ■

Proof There is always a solution in the game GCk . Actions of agents.

xi = 0, Vi G N

Values of agents' utilities.

fi =0, Vi G N

□

Nonzero Nash Equilibrium for GCk If

then there is one more solution. Theorem 2. Let

E J >1

- r, jeN j

Er^ > i-

^2 - r,

— I n

jeN j

The larger threshold the larger strategy of an agent in the game GCk . Proof Strategies of agents.

ri

A-

j-, Vi G N,

y^^ -1

^2 - r,

jeN j

Derivative.

dx = 2 r- > 0, Vi G N,

dA V" __1 ' '

^2 - r,

jeN j

r

If

E j >1

- Tj jeN j

then there is one more solution Theorem 3. Let

E j > i.

^2 - t j

jeN j

The larger threshold, the larger value of utility functions of agent in the game Gck ■ Proof. Values of agents' utilities.

A2 (1 - Ti)t

fi = ----rr-,e N

r, 2 - r,

-0 <2-

> 0, Vi e N

□

Thus if one wants to increase the utility of agents, they should increase the plans that these agents should try to exceed. This conclusion looks very reasonable, regardless of the model.

3.2. Players with communication and consensus

We can model this case by a game

GcB =<N,X,F(A),CB(A = Ao) >

There could be a communication between agents and they can communicate according the de Groot model (DeGroot, 1974). There is no difference if an existence of such communication to the common knowledge among all agents or it is not. Let their influences be wj then one should compose and solve the system

xi = BRi(x-j) = 12TT I ^ xj - wjAi 1 \i=j j

for each i.

Zero Nash Equilibrium for GCb There is always a zero solution in the game.

Theorem 4. An existance of a zero Nash equilibrium doesn't depend on value of the threshold in the game Gcb.

Proof. Strategies of agents.

xi =0, Vi e N

Values of agents' utilities.

fi = 0, Vi e N

Monotone Properties of Information Control in a Game with Uncertainty 117 Nonzero Nash Equilibrium for GCb If

then there is one more solution Theorem 5. Let

E

jew

E

jew

2 - r

> 1

2 - r

> 1.

If an agent in the game GCb has a nonzero influence on consensus opinion, then the larger his belief about the threshold, the larger the strategies of all agents. The true value of the threshold doesn't affect the strategies in the GCb .

Proof. Strategies of agents.

Derivative.

'2 - ri

Xi ---—-, Vi G N

E w Aj

jew

E

jew

r

j

1 - rj

-1

dx.

dA

j

2 - ri

j E j -1

1 - rj

jew j

> 0, Vi G N

dxi _ _

dAi _ E

jew

' 2 - ri

> 0, Vi G N

1 - rj

-1

dxi

dA

_ 0, Vi G N

If

then there is one more solution Theorem 6. Let

y —j— > 1

^2 - r,

je w j

E

jew

2 - r

> 1.

□

If an agent has a nonzero influence on consensus opinion, then the larger his belief about the threshold, the larger the utility for each agent. The true value of the threshold doesn't affect the utilities for agents.

r

r

r

j

Proof. Values of agents' utilities.

Aj (1 - ri)ri

h'EN

fi = -r^-1-, Vi e N

£ 2-rj - M <2 - )2

IjEÄ j

Derivative.

wj E (1 - ri)ri

dfi jew

2

> 0, Vi e N, j = i

£ j - H (2 - r.)2

U'eN j

wi E wj Aj (1 - r.)r. f =-^-2-> 0, Vi e N

dA.

- M (2 - r.)2

2-r

f =0' Vi e N

□

3.3. Players without communication

We can model this case by games

= < N, X, F(A), ViBiCB(A = Ai) >

GviBiCk(A=Ai) =< N,X,F(A), ViBiCK(A = Ai) > We formulated axioms to make an informational system complete.

GviBiCB(A=Ai)/\Bi(A=Ai) =<N,X,F(A), Vi(BiCB(A = Ai) A Bi(A = Ai)) >

GviBiCK(A=Ai)/Bi(A=Ai) =< N, X, F(A), Vi(BiCK(A = Ai) A Bi(A = Ai)) > i

fi = xi() ] x j — Ai) — xi /ri •

jeN

It coincides with the Nash equilibrium with a certain value of parameter A, if there is A = Ai for any i common knowledge that A = Ai.

The strategy of each player which equals to their best response that are

2ri

xi = BRi(x-i) = i — ir (^Jxj — Ai).

i j=i

2

Agent i makes a best response for all other agents according to their beliefs. Thus, from the i-th player's point of view it looks like they should compose and solve the system for the following best responses

2r.

xj = BRi(x_i) = 1 _(^xfc - A,)

for each j.

Zero Nash Equilibrium for GViBiCb (A=Ai)ABj (A=Ai) There is always a solution.

Theorem 7. An existance of a zero Nash equilibrium doesn't depend on value of the threshold in the game GViBiCK(A=Ai)ABi(A=Ai)-

Proof. Strategies of agents.

x, = 0, Vi e N

Values of agents' utilities.

fi =0, Vi e N

□

Nonzero Nash Equilibrium for GviBiCB (A=Ai)ABi(A=Ai) If

y^^ > i

^2 _ r ,

jew j

then there is one more solution. Theorem 8. Let

^2 _ r

E j > i.

^2 _ r,

jew j

One can change a strategy of an agent in the game GViBiCk(A=Ai)ABi(A=Ai) =< N, X, F(A), Vi(BiCK(A = Ai) A Bi(A = Ai)) if and only if she change his belief about the threshold. In this case the larger belief about threshold will lead to the larger strategy.

Proof. Strategies of agents.

A,-- ri

—, Vi e N

V _ 1

2_r

2_r

je w

Derivative.

dx, 2 _ r, „ w. ,r

■mr = ^—-> 0, Vi e N,

dA, V^ _ i

jew j

=0, Vi e N, j = i. dx,

dA = 0'Vi e N

x

r

Thus one cannot change a strategy of an agent if she doesn't change his own belief about A.

Theorem 9. Let

E

jew

2 - r

> 1.

The utility of an agent in the game GViBiCB(A=Ai)ABi(A=Ai) will increase when his belief A will increase if and only if this statement holds

2A < V a., j - A f V j - 1

12 - n j-i j 2 - rj

2-

Uew

Proof. Values of agents' utilities.

2 - n

Ai£ Aj j - ^lE - 0 - A2

£

I jew ■

2 - ri

-1

jew

Uew

II

2 - ri

2 - ri

Derivative

Vi G N

dfi

dA

Ai-

' 2 - ri

> 0, Vi G N j = i

V - 1

^2 - rj

jew j

-2

f = A.A__nrj_I V -T

dAi i j (2 - ri) (2 - rj) I 2w 2 - r,

E^^T - 1 > 0,Vi G Nj = i

dAi

2 - ri

Y^^ -1

^2 - r,

jew

A ri + V^ a rj - A V^ rj - 1 I - 2A 1 i 2 - ri ^ j 2 - r, 1^2 - r, I i 2 - r.

jew

2-r

2 - rj

jew j

'2 - ri

2 - ri

^2 - r,- 1

Vjew j

EAj j - a(E

jew

jew

2 - t,

- 1 - Ai

j

2

r

r

r

x

2

x

r

r

j

2

It means that sometimes we may need a combined information control - no separately chosen beliefs about thresholds. One can get some details from this theorem.

Theorem 10. Let

^2 - r,

jew j

If this statement

E^j (A - A) > °>Vi e N

jew j

doesn't hold then the utility of all agents in the game GViBiCb(A=Ai)ABi(A=Ai) won't increase at the same time when all their beliefs Ai increase.

Proof. Let's list all inequalities

E a j - A(E j - ^ - A > Vi ^N

jew j Vjew j

Aj-^ - y -^¿L- - 1 ) - Aj——— > 0, Vi G N

2 - j 2 - —j 2 - — ^ 2 - —j / j 2 - — '

Y^^T A^-^^ - AV-^ (V - l) -V Aj——— > 0,

Z^ 2 - ^ m 2 - —m ^2 - l ^ 2 - —/ ^ j 2 - —, '

jeN j meN m jeN j \jeN j ) jeN j

Vi G N

E 5-j - ') (s Aj - A E j >0-Vi g N

jeN j jeN j jeN j

E j - >) E j(A» -A) >0Vi g N

je N j je N j

E^j (Aj - A) > 0 Vi G N

2 - rj

je N j

3.4. Stubborn players with communication without consensus

We can model this case by games

GBiCB(A=Ai) =< N,X,F(A), ViBiCB(A = Ai) >

GBiCK (A=Ai) =< N, X, F (A), ViBiCK (A = Ai) > We formulated axioms to make an informational system complete.

GbcB(A=Ai)ABi(A=Ai)) =<N,X,F(A), Vi(BiCB (A = Ai) A Bi(A = Ai)) >

GBiCK (A=Ai)ABi(A=Ai) =< N, X, F (A), Vi(BiCK (A = Ai) A Bi(A = Ai)) >

□

If there is a communication with no trust at all then all agents 'become stubborn" and other opinions donBT)™t change their opinions. There is no difference if an existence of such communication is a common knowledge among all agents or it is not. The important information is that A, is a common knowledge and that all agents are stubborn in our sense. Thus, from the i player's point of view, they should compose and solve the system for the following best responses

xj = BRi(x-i) = " | ^ ] x j — 1 - ri \j=i

i

Zero Nash Equilibrium for Gvi(BiCK (A=Ai)ABi(A=Ai)) There is always a solution for (A=Ai)ABi(A=Ai) and GBiCB(A=Ai)ABi(A=Ai))-

Theorem 11. An existance of a zero Nash equilibrium doesn't depend on value of the threshold in the games Gb^Ck(A=Ai)ABi(A=Ai) ««d Gbce(A=Ai )ab» (A=Ai)) •

Proof. Strategies of agents.

x, =0, Vi e N

Values of agents' utilities.

fi = 0, Vi e N

□

Nonzero Nash Equilibrium for Gyi(BiCK (A=Ai)ABi (A=Ai)) If

£ j > 1

icW

then there is one more solution. Theorem 12. Let

Then there is

jew

> i.

- r. jew j

> 0; Vi e Nj = i,

dx,

dA > 0

in the game Gv^Ck(a=a* )ab* (a=a* )) if and only if

A, < 1, Vi e N.

Proof. Strategies of agents.

2 - ri

V rj - 1 I ' ^2 - r 2-r

jew

Ai + E2j (Aj - Ai ) , Vi e N

2 - r,- * \ jew

Derivatives.

' j ' j

2 - — ^ 2 - —,

dxj _ j j=j j

dAj _ y- —j - 1 2-—

—. > 0, Vi G N, Vj _ i

j " j

jeN —j

j

dxj 2 - —

— |1 - A^y —j— , Vi G N

<9A, V^ —j 1 I j ^ 2 - I'

j 2-2 - - jeN j

je N

Theorem 13. Let

1 —j a > jeN j >1,

j jeN j

§A _0, Vi g N

> i.

2 - —j

jeN j

□

The larger the threshold A the smaller the value of the utility function of each agent in the game GVi№Ck{A=Ai)ABi{A=Ai))

Proof. Let's find an universal expression for derivatives of utility functions. If

dA =!

then

dfj _ - - dXi a dA _ Xj dA

Thus

—-

f _ - _ - 2 - —j dA Xj —j - 1 l"* ' ^2 - —

A- + E 5j (Aj - AjM < 0, Vi G N

Z-2T7J - 1 \ jeN

jeN j

Values of agents' utilities

fi(xi, —n, Ai, ...A„) _

□□

2 - —j 'A- (Aj - A-)

—j jeN

y -1

jeN j

yA£l(Y -1) + nYjAjlL - a| x

^2 - r. \ ^ 2 - rm / ^2 - r. I jew j Vmew m / jew j

r2

2-1 Ai + E 2-j (Aj- Ai}

x_(2_- ri) I A , ^ 'j

E^—-1 r

rj jew

jew

r

j

2 - ri T (Ai + E2j (Aj - Ai) je w 2 - rj

V ^^ -1

jew j

2

j £ j+"- 0 - A £ j - M -

jew j jew j jew j

Ai + E — (Aj - AiH , Vi e N

2 - r, ' ^ 2 - r.

i jew

Theorem 14. Let

If there are

and

jew j

> 0

2-A

x, < r,-

2

in the game Gy^BiCK{A=Ai)ABi{A=Ai)) then

Proof. Since x, > 0 then

If

dfi

IT > o.

dy

dfi — < 0.

dA <

£ = 0

dy

then

dfi dxi ^ dxj dxi dxi x,

- = - + / - — A- — 2- * —,

dy dy ^^ dy dy dy r,

2

x

2

1

dfj _ 2dXi (1 - A - Xi^l + V^ dxj

dy dy * 2 — I dy '

j=j

dy

□□

Theorem 15. Lei

E

jeN

2 - —,

> 1.

Then

fi _ dAj

2 - 2 - —,

j j=i j

y ^^ -1

jeN j

A 1 2 - — 1 - — -

\

—k

2 T-y^ __1

2-—

E

jeN

—m

A- + E2j (A - A- ) je N 2 - —j

j

+

2 - —k 2 - —m

Em=k

- ^ =-+

2 - —,

k=i,j

E1 m j x ~ — m 1

2 - —m 2 -

1 - A^E

me N

meN

2 - —m

me N

2 - —m

in the game Gyj№Ck{A=Ai)ABi{A=Ai))-Theorem 16. Let

Then there is

y^^ > 1.

jeN j

f _ 2 2 - —, dAj

E

jeN /

2 - —j

a1 j 1 x

- 1 \ jeN j

1 - A - 1 2 - —j

—

j

2 —j y

2 - —j

jeN j

-1

A- + E2j (A' - A-) jeN 2 - —j

- — , 2 - '

+

2 —j = . 2 —m

^ m—m—, Vi g N, Vj _ i

j=j

E

meN

2 - —m

-1

—

j

—

—

j

—

—

in the game gy-^Ck(a=aî)abî(a=aî))-

4. Conclusion

In this paper we considered a game with a parameter A and suggested that this is an uncertain parameter for agents and they have to make some suggestion about it. We made a table where listed behavior of derivative of strategies and utilities of agents. Having this table one can predict the reaction of a system and thus choose appropriate informational control.

Table 1. PI ease write your table caption here

Control Game StrategjUtility

xi Ji

d/dA Players with common knowledge GCk > 0 >0

Players with communication and consensus GCb = 0 = 0

Players without communication GViBi cB (a= = 0 >0

Stubborn players with communication = 0 < 0

GVi(bick (A=Ai)ABi(A=Ai))

d/dAj Players with common knowledge GCk NA NA

Players with communication and consensus GCb >0 >0

Players without communication GViBi Cb (A= >0 ?

Stubborn players with communication ? ??

Gvt(BiCK (a = ai)Abi(a=ai))

d/dAi Players with common knowledge GCk NA NA

Players with communication and consensus GCb >0 >0

Players without communication GViBi Cb (A= >0 >0

Stubborn players with communication > 0 ??

g^i(bick (a=ai)abi(a=ai))

5. Acknowledgement

The article was prepared within the framework of the HSE University Basic Research Program and funded by the Russian Academic Excellence Project '5-100

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