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Information, Control and Games

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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Information, Control and Games

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  1. 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

  2. Moral Hazard, Incentives Theory (continued), and Incomplete Information

  3. 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

  4. 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?

  5. 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>w2u’’(w)<0 ==> u’(w1)<u’(w2) if w1>w2

  6. 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)

  7. 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

  8. 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

  9. 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

  10. 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 = w2EU=(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?

  11. 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」

  12. 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」, di.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

  13. 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

  14. 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)

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. Equilibrium under incomplete info. • Joint solution a-c- b Q2() = 2b Q1a-c-  - b Q1 = 2b Q2 • Output in equilibrium • Expecting E() =0, firm 1 produces as usuallyQ1* =(a-c)/3b • Known , firm 2 producesQ2*()= (a-c)/3b -  / 2b • Price in equilibrium • P*() = a-b[Q1*+Q2*()] = a-b[Q1*+Q2*] + /2, orP*() = P* + /2 (note: P*=P(Q1*+Q2*))

  23. 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)

  24. If  is also known to firm 1 • informed Firm 1 • max 1 = [a -c- b (Q1+ Q2()) ]Q1 • FOCa-c- b Q2() = 2b Q1 () • Firm 2 • max 2 = [a -c-  -b (Q1+Q2) ]Q2 • FOCa-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)

  25. 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 )

  26. Profit 比較 • Output in equilibrium for unknown  • Firm 1 • 1 = [P*+ /2-c][(a-c)/3b] • Firm 2 • 1 = [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 2 • 2**() = [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

  27. 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.

  28. 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)

  29. 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 )

  30. 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

  31. 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

  32. 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)

  33. 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, ^ → 

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