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Information, Control and Games

Shi-Chung Chang

EE-II 245, Tel: 2363-5251 ext. 245

scchang@cc.ee.ntu.edu.tw, http://recipe.ee.ntu.edu.tw/scc.htm

Office Hours: Mon/Wed 1:00-2:00 pm or by appointment

Yi-Nung Yang

(03 ) 2655205, yinyang@ms17.hinet.net

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Moral Hazard, Incentives Theory (continued), and Incomplete

Information

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Moral hazard

• 道德風險– A person who has insurance coverage will have

less incentive to take proper care of an insured object than a person who does not

• Two players are involved:– Insurer (manager of the insurance company)– Customer of the insurance company

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The essential question in incentive scheme design

• The essential question:– What kind of insurance will the customer buy?

– Coverage v.s. carefulness

– Moral hazard problem, Adverse selection, and its cures

• How to formulate the problem mathematically?

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Insurance market

• 假設 & 定義– 原始財富水準 w

– 發生意外機率 , 損失 L

– 為防止意外繳交保費 ,

– 投保額 z ( 即發生意外之後 , 投保人獲償之金額 )

– q 為每單位投保額所需繳交之保費 ( 由保險公司決定 )( 故選擇投保額 z 者 , 需繳交保費 = qz)

– 消費者效用函數 = u(w)• 風險趨避的假設隱含 :

u(w) in increasing in w but at a decreasing rate, i.e., u’(w)>0 ==> u(w1)>u(w2) if w1>w2

u’’(w)<0 ==> u’(w1)<u’(w2) if w1>w2

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投保者 ( 消費者 ) 的期望效用極大化

• 投保者 ( 消費者 ) 選擇 z, 以尋求期望效用最大 :即求解 :max (1- )u(w-qz) + u(w-qz-L+z)– 令 w1= w-qz, w2= w-qz-L+z

– 上式對 z 偏微分求解最適投保額 z , 其一階條件為 (1- ) u(w1)(-1)q+ u(w2)(-q+1)=0, 或

(1- ) u(w1) q= u(w2)(1-q)

• 再假設保險公司收到的保費剛好用來支付理賠– qz= z (q= ), 代入上式 , ( 如果 u(.) is a monotonic

function) 可得 : u(w1)= u(w2) ==> w-qz= w-qz-L+z==> z = L ( 消費者全額投保 )

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Moral Hazard ( 道德風險 )

• 若個人發生意外的機率 與其小心程度 x 有關 = (x), for x ≥0, 且– 愈小心的人 , 發生意外的機率愈低 , i.e.,

(x)/x = (x) <0

• 若保險公司無法觀察每人投保人之「小心程度」– 而將每單位保費設為相同的 q,

– 則投保人 ( 消費者 ) 尋求期望效用最大時 , 同時選 z 和 x, 即求解 :max EU=(1- (x))u(w-x-qz) + (x)u(w-x-qz-L+z)

– 其一階條件為 ( 令 w1= w -x -qz, w2= w -x -qz-L+z )(1) EU /z=(1- (x)) u(w1)(-q)+(x) u(w2) (1-q) =0

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保險為完全競爭市場下之道德風險 (1/3)

• 第 1 個 FOC (1- (x)) u(w1)q=(x) u(w2) (1-q)

• 假設保險公司收到的保費剛好用來支付理賠 , i.e., q= (x)– 代入 (1) 式得 :

(1- (x)) u(w1) (x)=(x) u(w2) (1-(x)) ==> u(w1)=u(w2)==> w1=w2 ==> w-qz= w-qz-L+z ==> z = L

– 消費者會全額投保• 使得其無意外之所得水準 w1= w2 發生意外之水準

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保險為完全競爭市場下之道德風險 (2/3)

• 第 2 個 FOC EU /x = -(1- )u(w1)- u(w1)- u(w2)+ u(w2)

=0??

– 代入第 1 個 FOC 的結果 (w1=w2)

EU /x = -(1- )u(w1)- u(w1)- u(w1)+ u(w1)

= u(w1)(-1+ - ) = -u(w1)<0 (recall u(w)<0)

EU /x <0implies that x 愈小 ==> EU 愈大

– x = 0 ==> 投保者小心程度 = 0

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保險為完全競爭市場下之道德風險 (3/3)

• 檢視目標函數max EU=(1- (x))u(w1) + (x)u(w2)

– 若 EU /z= 0 成立 , 則 w1 = w2

– No matter what the outcome will be, the insured person with a full coverage gets the same level of u(w).

• Implications for w1 = w2 EU=(1- (x))u(w-x-qz) + (x)u(w-x-qz-L+z)– The level of x determined by the person do affect the outcome

(probability), but ...

– The ultimate utility levels (as well as income) are the same.

• 如果你是消費者 , what will you do?– 全額投保 ?

– 小心保管 ( 使用 ) 你的投保物品 ?

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Market designs for insurance

• 從數學求解的觀點– The problem (of moral hazard) is caused by the 2nd FOC:

EU/x = -(1- )u(w1)- u(w1)- u(w2)+ u(w2) 0 = -u(w1) < 0

– because w1=w2 (so this problem is originally from 1st FOC)

– so, there is a corner solution, i.e., x=0 (recall that = (x), for x ≥0, x 是小心程度 )

– If we can do something to allow what could happen: EU/x =0, ... Or w1 w2 ==> z L

– 不能讓消費者選「 full coverage 」

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Deductibles as a Mechanism

• 若保險公司政策是 : 「不能讓客戶買全險」 ...

• Deductibles 自付額– z =L 是保額 , 但理賠時需負擔「 deductible 」 , d

i.e., 意外時賠 z-d

– w1 = w-x-qz, w2= w-qz-L+z-dso that w1 > w2

• The 2nd FOC: EU/x = -(1- )u(w1)- u(w1)- u(w2 )+ u(w2) = -u(w1)+ [u(w1)-u(w2)]+ [u(w2)- u(w1)] (-) (+) (-) (-) (-)

• 有可能 EU/x =0– 所以 x 0

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Economic Insight of the Deductibles

• 檢視目標函數max EU=(1- (x))u(w1) + (x)u(w2)

• Incentive– w1 > w2– if the consumer can increase x to reduce (x),

– this gives more weights on (1- ) u(w1). So, he will be more careful (x↑)

• Insurance policy– d 愈大 , 則 w2 愈小 ==> x↑

防止汽車被偷 , 記得鎖車門 , 加買大鎖 , 裝 GPS 防盜 ....– 保費可能也不同

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Self-selection Condition

EUI EUU

• 未投保 ( 自己小心 )– EUU=(1- (x))u(w-x) + (x)u(w-x-L)

• 全額投保 with deductibles– EUI=(1- (x))u(w1) + (x)u(w2)

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Adverse Selection

• 逆向選擇– 小心的消費者不投保 , 粗心的消費者都來投保

• 從數學求解的觀點– w1=w2 (from 1st FOC) is because q = (x)

保險公司收支平衡 ,

– q 是平均保費• 兩種消費者小心程度不同

xH> xL ==> (xH)<(xL)

• 保費相同時 q = (1/2) [(xH)+(xL)](xH)<q<(xL)

– Adverse selection• 小心的消費者覺得保費太貴 , ... 可能不加入保險• 粗心的消費者覺得保費很合算 , ... 全部加入保險

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Incomplete Information in a Cournot Duopoly

• Complete information– A player knows

• who are the other players

• what are their strategies

• what are their preferences ...

• Incomplete information– A player is unsure about the answer to some or all of the

above question

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Basic model of a Duopoly market

• Players: Two firms 1and 2 Identical products Output: Q1 and Q2 Same constant marginal costs: c

Total cost = a Qi

• Market (inverse) demandP = a - b Qwhere Q = Q1 + Q2 , a, b>0

• Complete information– firm i: max. i = P(Q)Qi - cQi = (P-c) Qi

– FOCP Qi + P-c =0

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Solutions of the Basic Dupoly market

• firm 1: max. 1 = P(Q1+Q2)Q1 - cQ1 = (P-c) Q1 = [a -c- b (Q1+Q2) ]Q1

– FOC (response function) [a -c- b (Q1+Q2) ] -bQ1 = 0==> a -c- bQ2 = 2bQ1

• firm 2: max. 2 = P(Q1+Q2)Q2 - cQ2 = [a -c- b (Q1+Q2) ]Q2

– FOC (response function) ==> a -c- bQ1 = 2bQ2

• 聯立求解Q1* = Q2* = (a-c)/3bP* = (a+2c)/3

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Scenario of incomplete information

• Firm 2’s costs are unknown to firm1but firm 1’s costs are known to both players

• Firm 2 has a constant marginal cost = c + where e (, ) with a prob. dist. F, E() = 0– firm 2 has cost advantage if e<0 is known to firm 2 but not to firm 1

– but F is known to both firms

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Profit max. under incomplete info.

• Firm 2– given conjecture that firm 1 produces Q1

– max 2 = [a -c- -b (Q1+Q2) ]Q2

– FOCa-c- - b Q1 = 2b Q2

– response function of firm 2Q2 = (a-c- - b Q1 )/2b if Q1 (a-c- )/b = 0 if Q1 > (a-c- )/b

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Profit max. under incomplete info.

• Uninformed Firm 1– He knows different types () of firm 2 will produce different

Q2.

– He expects output of firm 2 =EQ2() = Q2

– given conjecture that firm 2 produces EQ2()

– max 1 = [a -c- b (Q1+ EQ2()) ]Q1

– FOCa-c- b Q2() = 2b Q1

– response function of firm 2Q1 = (a-c- b Q2() )/2b if Q2() (a-c)/b = 0 if Q2() > (a-c)/b

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Equilibrium under incomplete info.

• Joint solution a-c- b Q2() = 2b Q1 a-c- - b Q1 = 2b Q2

• Output in equilibrium– Expecting E() =0, firm 1 produces as usually

Q1* =(a-c)/3b

– Known , firm 2 producesQ2*()= (a-c)/3b - / 2b

• Price in equilibrium– P*() = a-b[Q1*+Q2*()] = a-b[Q1*+Q2*] + /2, or

P*() = P* + /2 (note: P*=P(Q1*+Q2*))

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Profit in Equilibrium under incomplete info.

• Firm 1 1 = [P*()-c ]Q1*

= [P*+ /2-c] Q1* = [P*+ /2-c][(a-c)/3b]

• Firm 2 1 = [P*()-c ]Q2*()

= [P*+ /2-c- ] [Q1*- /2b] = [P*- /2-c] [Q1*- /2b]

>0, firm 2 相對成本較高 ( 相對於 =0) 1 較大 2 較小 ([P*- /2-c] 且 [Q1*- /2b] 皆較小 )

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If is also known to firm 1

• informed Firm 1– max 1 = [a -c- b (Q1+ Q2()) ]Q1 – FOC

a-c- b Q2() = 2b Q1 ()

• Firm 2– max 2 = [a -c- -b (Q1+Q2) ]Q2 – FOC

a-c- - b Q1 () = 2b Q () 2

• Equilibrium output (with complete info. about )– Q1**() = (a-c)/3b + /3b– Q2**() = (a-c)/3b - 2/3b

• Equilibrium price– P**() =P* + /3 (recall P*= (a+2c)/3)

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Output 比較

• Output in equilibrium for unknown – Q1* =(a-c)/3b

– Q2*()= (a-c)/3b - / 2b

• Output in equilibrium (with complete info. about )– Q1**() = (a-c)/3b + /3b

– Q2**() = (a-c)/3b - 2/3b

> 0 ( 反之 , 同理可推 )– firm 1 產量較多 ( 因為確定 firm 2 成本較高 )

– firm 2 產量較小 ( 因為知道 firm 1 在知道 > 0, 產量較大 )

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Profit 比較• Output in equilibrium for unknown

– Firm 11 = [P*+ /2-c][(a-c)/3b]

– Firm 21 = [P*- /2-c] [Q1*- /2b]

• Profit in equilibrium If is also known to firm 1– Firm 1

1**() = [P* + /3-c ] Q1**() = [P* + /3 - c ][(a-c)/(3b) + /(3b)]

– Firm 22**() = [P* + /3-c- ] Q2**()

= [P* - (2 )/3-c][(a-c)/3b - (2)/(3b)]

• Incentive for firm 2 to reveal its to the public if <0– firm 2 的利潤較大 if <0 is also known to firm 1

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Conjecture

• A low-cost firm 2 benefits from having its cots made public– because the consequent price s higher and it produces more

in equilibrium.

• Conversely, a high-cost firm 2 suffers – because it sells a smaller quantity at a lower price.

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Revealing Costs to a Rival

• An efficient firm 2 (<0) will make the information about its low costs public.

• Q: How about an inefficient firm 2 with >0?

• Reasoning– Efficient firms will reveal their costs to its rival.

– But non-revelation is also informative:不願透露成本訊息的廠商 , 很有可能是高成本 (>0)

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Informative no-information

• In 1st stage

• Firm 2 can decide to reveal or not reveal – assumptions: information revealed by firm 2 is credible and

costless.

• In 2nd stage

• After revelation or lac thereof- the two firms compete on quantities

• Focus on:– Firm 1 concludes from non-revelation that

firm 2’s costs must be higher than some level

– ( > >^ >0 )

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All type of the firm reveals their costs

• Proposition– In equilibrium, =^ , every type of firm 2 will reveal its

costs

• Thinking: – for any ^ < and non-revelation about firm 2’s costs

– firm 1 可假定 firm 2’s type 介於 (^, ) 之間進而猜測其產量為 Q~

2

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Proof for the Proposition (1/2)

• FOCs– Firm 1 ( 令其預期 E() = - , 然後當已知條件 )

a-c-bQ~2 = 2bQ ~

1

– Firm 2 ( 也了解未透露 的可能後果 ) a-c- -bQ~

1 = 2bQ ~2

• Output in equilibrium for non-revelation– Q ~

1 = Q1* + (-)/3b

– Q ~2() = Q2* - [(-)/(6b) + /(2b) ]

recall Q1* = Q2*= (a-c)/3b

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Proof for the Proposition (2/2)

• Price in equilibrium for non-revelation– Q~

1 = Q1* + (-)/3b

– Q~2() = Q2* - [(-)/(6b) + /(2b) ]

P~ = a-b[Q~1+ Q~

2 ()] = a-b[Q1* + (-)/3b + Q2* - (-)/(6b) - /(2b)] = P* + /2- (-)/6 (recall P*=P(Q1*+ Q2*)

• Price in equilibrium for non-revelation– Firm 1

~1= (P* + /2- (-)/6 -c)[Q1* + (-)/(3b)]

– Firm 2~

2= [P* + /2- (-)/6 -c- ][Q2* - (-)/(6b) -/(2b) ] = [P* - /2- (-)/6 -c] [Q2* - (-)/(6b) -/(2b) ]

– Firm 2 suffers when - > ^ (compared to true is known)

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Summary of non-revelation

• All types of firms would prefer to reveal their costs in the 1st stage– firms that have a cost between ^ and - prefer to reveal thei

costs rather than not reveal.

– Firm 2 observes cost information between ^ and -, and raises his guess about -, and so on ...

– In equilibrium, ^ →