Lorna Pamba & Karol Rakowski

A Game-theoretic approach to non-life insurance.

Introduction:

• Goal of research as mathematics majors with minors in economics

• Nash Equilibrium: a stable state of a system involving the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged

Some background Information; • Hawk and Dove Game

• Player choice either Hawk(H) or Dove (D)

• Four pure strategy combinations ; HH,HD,DH,HH

is the reproductive value of territory is the cost of being injured in a fight

In bird's 1 point of view, lets consider the cases:

• DH( Bird1 plays dove and Bird 2 plays Hawk)

Payoff = 0• HD( Bird 1 plays Hawk and Bird 2 plays Dove)

Payoff = ρ• DDExpected payoff=0*Prob(F=I)+ρ*Prob(F=II)=ρ/2

• HH

Exp=ρ*Prob(S=I)+(-C)*Prob(S=II)=(( ρ-C))/2

  H D

H 

D

  H D

H 

D

Payoff Matrices;Bird 1 Bird 2

Probabilities of mixed strategies;Bird I, play H with probability u and play D with probability

Bird II, play H with probability v and play D with probability If I and II choices don’t affect each other;

= =

  HH HD DH DDPossible values

 

 

0  

 

Probabilities

Then f1(u,v)= E(F1) = =

Then f2 (u,v)= E(F2 ) = =

Similarly, f2 (u,v) = E(F2 )

For a fixed v

For a fixed u

Therefore bird I’s reaction set is; R1 =

And bird II’s reaction set is; R2 = {

R1= blue lineR2= Green lineIntersection: Nash equilibrium

2 companiesFixed number of customers in marketPossible strategies (increase or decrease in premiums) Represented by i and d.Disclaimer of quantatative “real world” application of following work

Insurance Market

Variable Defined As:

I Insurance Company 1

II Insurance Company 2

rI Premium for I

rII Premium for II

xI Increase in Premium for I

xII Increase in Premium for II

yI Decrease in Premium for I

yII Decrease in Premium for II

T Total Number of Customers in Market

PI Number of Customers for I

PII Number of Customers for II

RI Revenue for I

RII Revenue for II

Definitions

Change in Revenue = (Change in Number of Customers) * (Change in Premium) - (Previous Revenue).

Change in Revenue = ((∆rI,II - ∆rII,I) * (.01 * T) + PI,II) * (rI,II + ∆rI,II) – ((PI,II) * (RI,II))

Payoff – Change in Revenue

  I’s Payoff

  I II

d

 

i

 

(((-xI + xII)*(.01T) + PI)*(rI + xI)) - RI

 

(((-xI - yII)*(.01T) + PI)*(rI + xI)) - RI

 

I  

(((yI + xII)*(.01T) + PI)*(rI - yI)) - RI

 

(((yI - yII)*(.01T) + PI)*(rI - yI)) - RI 

d

Payoff Matrices

  II’s Payoff

  I II

d

 

i

 

(((-xII - xI)*(.01T) + PII)*(rII + xII)) - RII

 

(((yII + xI)*(.01T) + PII)*(rII - yII)) - RII

 

I  

(((-xI - yI)*(.01T) + PII)*(rII + xII)) - RII

 

(((yII - yI)*(.01T) + PII)*(rII - yII)) - RII 

d

Payoff Matrices

4 possible outcomes for both companies

Want to create a payoff function so we can examine which strategy is best choice

dominant strategy (a strategy is dominant if it is always better than any other strategy)

Payoff Matrices

we need to establish the probabilities, for both companies, of determining their decision to increase or decrease their premiums. Let: u = ( Prob{I}=i ) and v = ( Prob{II}=i ) where 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1 1 – u = ( Prob{I}=d ) and 1 – v = ( Prob{II}=d )

Constructing Payoff Functions

Constructing Payoff Functions

Company 1 ( I )

Probabilities

Strategies

Possible Outcomes (Payoffs)

uv

ii

(((-xI + xII)*(.01T) + PI)*(rI + xI)) - RI

u(1-v)

id

(((-xI - yII)*(.01T) + PI)*(rI + xI)) - RI

(1-u)v

di

(((yI + xII)*(.01T) + PI)*(rI - yI)) - RI

(1-u)(1-v)

dd

(((yI - yII)*(.01T) + PI)*(rI - yI)) - RI

Constructing Payoff Functions

Company 2 ( II )

Probabilities

Strategies

Possible Outcomes (Payoffs)

uv

ii

(((-xII - xI)*(.01T) + PII)*(rII + xII)) - RII

u(1-v)

id

(((yII + xI)*(.01T) + PII)*(rII - yII)) - RII

(1-u)v

di

(((-xI - yI)*(.01T) + PII)*(rII + xII)) - RII

(1-u)(1-v)

dd

(((yII - yI)*(.01T) + PII)*(rII - yII)) - RII

ƒI(u,v) = ((((-xI + xII)*(.01T) + PI)*(rI + xI)) - RI)(uv) + ((((-xI - yII)*(.01T) + PI)*(rI + xI)) - RI)(1-v)u + ( (((yI + xII)*(.01T) + PI)*(rI - yI)) - RI)(u-1)v + ((((yI - yII)*(.01T) + PI)*(rI - yI)) - RI)(1-u)(1-v)

ƒI(u,v) = u(v((xI+rI)(-.01xIT + .01xIIT + PI) – RI) + (1-v)((xI+rI)(-.01xIT - .01yIIT + PI) – RI) + (v((rI-yI)(.01T(xII + yI) + PI) – RI) – ((1-v)((rI – yI)(.01yIT - .01yIIT + PI) – RI))) – v((rI-yI)(.01T(xII + yI) + PI) – RI) + ((1-v)((rI – yI)(.01yIT - .01yIIT + PI) – RI)))

Evaluation of Payoff Function (1)

⍵ = (v((xI+rI)(-.01xIT + .01xIIT + PI) – RI) + (1-v)((xI+rI)(-.01xIT - .01yIIT + PI) – RI) + (v((rI-yI)(.01T(xII + yI) + PI) – RI) – ((1-v)((rI – yI)(.01yIT - .01yIIT + PI) – RI)))

ƒI(u,v) = u⍵– v((rI-yI)(.01T(xII + yI) + PI) – RI) + ((1-v)((rI – yI)(.01yIT - .01yIIT + PI) – RI)))

Evaluation of Payoff Function (1) cont.

ƒII(u,v) = ((((-xII + xI)*(.01T) + PII)*(rII + xII)) - RII)(uv) + ((((-xII - yI)*(.01T) + PII)*(rII + xII)) - RII)(1-v)u + ( (((yII + xI)*(.01T) + PII)*(rII - yII)) - RII)(u-1)v + ((((yII - yI)*(.01T) + PII)*(rII - yII)) - RII)(1-u)(1-v)

ƒII(u,v) = v(u((xII+rII)(-.01xIIT + .01xIT + PII) – RII) - u((xII+rII)(-.01xIIT - .01yIT + PII) – RII) + (u-1)((rII-yII)(.01T(xI + yII) + PII) – RII) – ((1-u)((rII – yII)(.01yIIT - .01yIT + PII) – RII))) + u((rII-yII)(.01T(xI + yII) + PII) – RII) + ((1-u)((rII – yII)(.01yIIT - .01yIT + PII) – RII)))

Evaluation of Payoff Function (2)

⍺ = (u((xII+rII)(-.01xIIT + .01xIT + PII) – RII) - u((xII+rII)(-.01xIIT - .01yIT + PII) – RII) + (u-1)((rII-yII)(.01T(xI + yII) + PII) – RII) – ((1-u)((rII – yII)(.01yIIT - .01yIT + PII) – RII)))

ƒII(u,v) = v⍺+ u((rII-yII)(.01T(xI + yII) + PII) – RII) + ((1-u)((rII – yII)(.01yIIT - .01yIT + PII) – RII)))

Evaluation of Payoff Function (2) cont.

Company 1 has no control over the value of v and hence has no control over the value of the expression v((rI-yI)(.01T(xII + yI) + PI) – RI) + ((1-v)((rI – yI)(.01yIT - .01yIIT + PI) – RI)))

The goal is to maximize u⍵ (when we maximize u⍵, we maximize ƒI). Again, we also know 0 ≤ u ≤ 1. So, when ⍵ is positive, negative, or equal to 0, we will have three different optimal values for u

Evaluation of Payoff Functions

If, ⍵ > 0 Then, u = 1If, ⍵ = 0 Then, All u ∈ [0,1]If, ⍵ < 0 Then, u = 0

Relation Between ⍵ and u

If, ⍺ > 0 Then, v = 1If, ⍺ = 0 Then, All v ∈ [0,1]If, ⍺ < 0 Then, v = 0

Relation Between ⍺ and v

A closer look at what determines the values of ⍵ and ⍺

The ⍵ and ⍺ equalities are crucial to examine as they have a direct impact on the u, v strategies that will be taken by Companies 1 and 2.

Further Evaluation

⍵ = (v((xI+rI)(-.01xIT + .01xIIT + PI) – RI) + (1-v)((xI+rI)(-.01xIT - .01yIIT + PI) – RI) + (v((rI-yI)(.01T(xII + yI) + PI) – RI) – ((1-v)((rI – yI)(.01yIT - .01yIIT + PI) – RI)))

⍵ = v (-2 RI + (xI + rI) (PI - 0.01 xI T + 0.01 xII T) + (-yI + rI) (PI + 0.01 (xII + yI)T) - (xI + rI) (PI - 0.01 xI T - 0.01 yII T) + (-yI + rI) (PI + 0.01 yI T - 0.01 yII T))+ (xI + rI) (PI - 0.01 xI T - 0.01 yII T) - (-yI + rI) (PI + 0.01 yI T - 0.01 yII T)

Evaluation of ⍵

δ = (-2 RI + (xI + rI) (PI - 0.01 xI T + 0.01 xII T) + (-yI + rI) (PI + 0.01 (xII + yI)T) - (xI + rI) (PI - 0.01 xI T - 0.01 yII T) + (-yI + rI) (PI + 0.01 yI T - 0.01 yII T))

a = (xI + rI) (PI - 0.01 xI T - 0.01 yII T) - (-yI + rI) (PI + 0.01 yI T - 0.01 yII T)

⍵ = v δ + a

Evaluation of ⍵ cont.

⍺ = (u((xII+rII)(-.01xIIT + .01xIT + PII) – RII) - u((xII+rII)(-.01xIIT - .01yIT + PII) – RII) + (u-1)((rII-yII)(.01T(xI + yII) + PII) – RII) – ((1-u)((rII – yII)(.01yIIT - .01yIT + PII) – RII)))

⍺ = u(-2 RII + (xII + rII) (PII - 0.01 xII T + 0.01 xI T) + (-yII + rII) (PII + 0.01 (xI + yII)T) - (xII + rII) (PII - 0.01 xII T - 0.01 yI T) + (-yII + rII) (PII + 0.01 yII T - 0.01 yI T))+(xII + rII) (PII - 0.01 xII T - 0.01 yI T) - (-yII + rII) (PII + 0.01 yII T - 0.01 yI T)

Evaluation of ⍺

λ = (-2 RII + (xII + rII) (PII - 0.01 xII T + 0.01 xI T) + (-yII + rII) (PII + 0.01 (xI + yII)T) - (xII + rII) (PII - 0.01 xII T - 0.01 yI T) + (-yII + rII) (PII + 0.01 yII T - 0.01 yI T))

b = (xII + rII) (PII - 0.01 xII T - 0.01 yI T) - (-yII + rII) (PII + 0.01 yII T - 0.01 yI T)

⍺ = u λ + b

Evaluation of ⍺ cont.

v = 1 ⍵ = δ + av = 0 ⍵ = av = V0 ⍵ = 0

Relation between v and ⍵

u = 1 ⍺ = λ + bu = 0 ⍺ = bu = U0 ⍺ = 0

Relation between u and ⍺

for every v which Company 1 has no control over, there exists a corresponding u which is a strategy that will make Company 1’s payoff/benefit as large as possible given the situation. So a rational reaction set is a set of all of these possible combinations, given an opposing strategy that a player has no control over

Rational Reaction Set

ZI = { (u,v) | 0 ≤ u, v ≤ 1 , ƒI(u,v) = max0 ≤ ū ≤ 1 ƒI(ū,v) }For each (u,v) in ZI, if Company 2 selects v then a best reply for Company 1 is to select u (a best reply rather than the best reply because there may be more than one). Note that ZI is obtained in practice by holding v constant and maximizing ƒI as a function of a single variable (whose maximum will depend on v). ZII = { (u,v) | 0 ≤ u, v ≤ 1 , ƒII(u,v) = max0 ≤ ῡ ≤ 1 ƒI(u,ῡ) } *Similar Explanation*

Rational Reaction Set

ZII - Blue Line

V0 ZI - Red Line

1

Nash Equilibrium 3

U0

Nash Equilibrium 1

Nash Equilibrium 2

Rational Reaction Sets

u

v

1

0

0

First Example

1

V0

U0

0.7

Rational Reaction Sets

u

v

1

0

0

ZII - Blue Line

0

0

Nash Equilibrium 2

Nash Equilibrium 1

Nash Equilibrium 3

ZI - Red Line

Example 2

ZII - Blue Line

V0

ZI - Red Line

1

Nash Equilibrium

Rational Reaction Sets

u

v

1

0

0

U0

Third Example

Rational Introspection: A NE (Nash Equilibrium) seems a reasonable way to play a game because my beliefs of what other players do are consistent with them being rational. This is a good explanation for explaining NE in games with a unique NE. However, it is less compelling for games with multiple NE. Markus Mobius, Lecture IV: Nash Equilibrium II - Multiple Equilibria. (lecture., Harvard, 2008), http://isites.harvard.edu/fs/docs/icb.topic449892.files/lecture42.pdf.Risk Aversion method in decision

Multiple Nash Equilibriums

A continuous Non-coperative game.

• 2 Insurance Companies; Geico and Progressive

• Battleground for the two companies;

Based on;

Area under the curve should be 1 ;

Since we are trying to estimate some realistic situation , we’ll say; Therefore our probability distribution function;

This is our pdf if the mean is chosen to be 13000.

10 00 0 20 00 0 30 00 0 40 00 0 50 00 0

0. 000 05

0. 000 10

0. 000 15

0. 000 20

4m̂ xe ^ 2xm

X axis represents the number of miles driven in a yearY axis represents the population in (100000000)

Some assumptions made

• Homogenous product but different level of satisfaction;

• Best premium rate for the Geico depends upon Progressive premium rate and vice versa.

• Geico and Progressive do not communicate with one another

• Utility is measurable.

• Let player 1 be Geico and player 2 be Progressive.

• Let be the premium rate set by Geico and be the premium set by Progressive

• Let u be the customers’ utilityAccordingly, we can say ans makes progressive the better choice.Similarly, we can say and makes Geico the better choice.

𝒖=𝒌𝒎𝒑𝟏−𝒌𝟏𝒎<𝒑𝟐−𝒌𝟐𝒎

Then we have 3 cases,

𝒌𝟐<𝒌𝟏𝒐𝒓𝒎>𝒑𝟐−𝒑𝟏𝒌𝟐−𝒌𝟏 𝑻𝒉𝒆𝒏𝒄𝒖𝒔𝒕𝒐𝒎𝒆𝒓𝒔 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆𝒆𝒏𝒕𝒊𝒓𝒆𝒍𝒚 𝒇𝒓𝒐𝒎𝑷

𝒌𝟐=𝒌𝟏 𝒊𝒎𝒑𝒍𝒊𝒆𝒔 𝒕𝒉𝒂𝒕 𝒑𝟐>𝒑𝟏

Without loss of generality, let’s assume all through that; ,

𝑮 (𝒔 )=𝒑𝒓𝒐𝒃 (𝒎≤ 𝒔 )= 𝟒𝐦𝛍𝟐 𝒙 ⅇ

−𝟐𝐱 /𝐦𝛍𝟎≤𝒎≤𝟏𝟑 ,𝟎𝟎𝟎

Let denote a cumulative distribution function

Let F1 denote Geico’ s payoff function;

Similarly progressive’s payoff function;

Let be Geico’ s and Nan’s strategy simultaneously,

𝒇 𝟏 (𝒙 ,𝒚 )=𝑬 ( 𝑭𝟏 )=𝒑𝟏 .𝒑𝒓𝒐𝒃 (𝒎≤ 𝒔 )+𝟎 .𝑷𝒓𝒐𝒃 (𝒎> 𝒔 )

=

=

And progressive’s reward function is

𝒇 𝟐 (𝒙 ,𝒚 )=𝒑𝟐(𝟏+−𝟔𝟓𝟎𝟎+ⅇ

−𝐌𝐚𝐱 [𝟎 ,𝟓𝟎𝟎 (−𝒑𝟏+𝒑 𝟐)]𝟔𝟓𝟎𝟎 (𝟔𝟓𝟎𝟎+𝐌𝐚𝐱 [𝟎 ,𝟓𝟎𝟎(−𝒑𝟏+𝒑𝟐)])

𝟔𝟓𝟎𝟎 )

Recall, that if Similarly;

The set of all feasible strategy combinations

Or similarly; |

Restricted price

with constant R2 = with constant

Recall,

¿𝝏𝒑𝟏 𝒇 𝟏[𝒑𝟏 ,𝒑𝟐]

𝑺𝒐𝒍𝒗𝒆 [𝟒𝟐𝟐𝟓𝟎𝟎𝟎𝟎 −𝟔𝟓𝟎𝟎ⅇ

𝒑𝟏−𝒑 𝟐𝟔𝟓𝟎𝟎(𝒌𝟏−𝒌𝟐) (𝟔𝟓𝟎𝟎+−𝒑𝟏+𝒑𝟐

𝒌𝟏−𝒌𝟐 )

𝟒𝟐𝟐𝟓𝟎𝟎𝟎𝟎 +𝒑𝟏(𝟔𝟓𝟎𝟎ⅇ

𝒑𝟏−𝒑 𝟐𝟔𝟓𝟎𝟎(𝒌𝟏−𝒌𝟐)

𝒌𝟏−𝒌𝟐 −ⅇ

𝒑𝟏−𝒑𝟐𝟔𝟓𝟎𝟎(𝒌𝟏−𝒌𝟐 ) (𝟔𝟓𝟎𝟎+−𝒑𝟏+𝒑𝟐

𝒌𝟏−𝒌𝟐 )

𝒌𝟏−𝒌𝟐 )

𝟒𝟐𝟐𝟓𝟎𝟎𝟎𝟎 =¿𝟎 ,𝒑𝟏]

This would help us find points in R1 if exact solutions were possible (would give potential p1 values to choose for a fixed p2).

This would help us find points in if exact solutions were possible (would give potential values to choose for a fixed).For example

20 0 40 0 60 0 80 0 10 00 12 00

20 0

40 0

60 0

80 0

10 00

12 00

This shows R1 in blue and R2 in red when k1=15/1000 and k2=17/1000.

Nash Equilibrium

Weaknesses of research

Justifications

Conclusion

 Questions?