a game-theoretic approach to non-life insurance
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A Game-theoretic approach to non-life insurance. Lorna Pamba & Karol Rakowski. Introduction: . Goal of research as mathematics majors with minors in economics - PowerPoint PPT PresentationTRANSCRIPT
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Lorna Pamba & Karol Rakowski
A Game-theoretic approach to non-life insurance.
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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
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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
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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
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H D
H
D
H D
H
D
Payoff Matrices;Bird 1 Bird 2
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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
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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
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Therefore bird I’s reaction set is; R1 =
And bird II’s reaction set is; R2 = {
R1= blue lineR2= Green lineIntersection: Nash equilibrium
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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⍵ = (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.
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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)
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⍺ = (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.
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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
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If, ⍵ > 0 Then, u = 1If, ⍵ = 0 Then, All u ∈ [0,1]If, ⍵ < 0 Then, u = 0
Relation Between ⍵ and u
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If, ⍺ > 0 Then, v = 1If, ⍺ = 0 Then, All v ∈ [0,1]If, ⍺ < 0 Then, v = 0
Relation Between ⍺ and v
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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
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⍵ = (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 ⍵
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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.
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⍺ = (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 ⍺
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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.
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v = 1 ⍵ = δ + av = 0 ⍵ = av = V0 ⍵ = 0
Relation between v and ⍵
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u = 1 ⍺ = λ + bu = 0 ⍺ = bu = U0 ⍺ = 0
Relation between u and ⍺
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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
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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
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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
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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
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ZII - Blue Line
V0
ZI - Red Line
1
Nash Equilibrium
Rational Reaction Sets
u
v
1
0
0
U0
Third Example
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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
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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;
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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)
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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.
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• 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.
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𝒖=𝒌𝒎𝒑𝟏−𝒌𝟏𝒎<𝒑𝟐−𝒌𝟐𝒎
Then we have 3 cases,
𝒌𝟐<𝒌𝟏𝒐𝒓𝒎>𝒑𝟐−𝒑𝟏𝒌𝟐−𝒌𝟏 𝑻𝒉𝒆𝒏𝒄𝒖𝒔𝒕𝒐𝒎𝒆𝒓𝒔 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒆𝒆𝒏𝒕𝒊𝒓𝒆𝒍𝒚 𝒇𝒓𝒐𝒎𝑷
𝒌𝟐=𝒌𝟏 𝒊𝒎𝒑𝒍𝒊𝒆𝒔 𝒕𝒉𝒂𝒕 𝒑𝟐>𝒑𝟏
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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;
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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 )
=
=
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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;
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The set of all feasible strategy combinations
Or similarly; |
Restricted price
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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).
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This would help us find points in if exact solutions were possible (would give potential values to choose for a fixed).For example
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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
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Weaknesses of research
Justifications
Conclusion
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Questions?