4.2 Static Games under Incomplete Information Information

4.2 Static Games under Incomplete Information 4.2.1 Information
Perfect Information: At each move in the game the player with the move knows the full history of the play of the game thus far (p55, Gibbons). Each information set is a singleton (p49, Rasmusen). Players know where they are. No moves are simultaneous. The strongest requirement. Any game of incomplete or asymmetric information is also a game of imperfect information. (p50, Rasmusen). 둘 중 하나만 해당되도 비완벽정보 게임. 완전하고 대칭정보이어야 완벽정보, Asymmetric, incomplete  imperfect? YES, Imperfect  asymmetric, incomplete? NO (slide 3) Certain Information: No move by nature after any player moves. A game of certainty can be a game of perfect information if it has no simultaneous moves.

4.2 Static Games under Incomplete Information
4.2.1 Information (cont’d) Complete Information: Each player’s payoff function (the function that determines the player’s payoff from the combination of actions chosen by the player) is a common knowledge among all the players (p1, Gibbons). Nature does not move first, or her initial move is observed by every player (p49, Rasmusen) Symmetric Information: A player’s information set at 1) any node where he chooses an action, or 2) an end node, contains at least the same elements as the information sets of every other player (p49, Rasmusen) Usually, information asymmetry implies information incompleteness. However, there are exceptions. Ex) Card games in the next slide. Ex) Prisoner’s Dilemma: imperfect, complete, symmetric, certain information game.

4.2.1 Information (cont’d) All players lack info on his own card AND others’ cards as well (incomplete but symmetric) Classifying information Ex) Game of Poker. Betting after cards. A player has info on his own card But lacks info on others’ cards  Asymmetric and Incomplete *Korean beef market? Imperfect-Uncertain-Asymmetric-Incomplete information game.

∩ Incomplete Info is big
4.2 Static Games under Incomplete Information 4.2.1 Information (cont’d) Perfectness, Completeness and Symmetry of Information Perfect Info: 특별한 경우 Symmetric & Imperfect Complete & Imperfect Complete Info Symmetric Info Imperfect Info: 불완벽정보게임 Asymmetric Info Incomplete Info Asymmetric Info ∩ Incomplete Info is big  비대칭정보≒불완전정보 Info. Completeness Complete Incomplete Info. Symmetry Symmetric Perfect Imperfect Asymmetric

4.2.1 Information (cont’d) Bayesian Games when ∃ info. asym. or none. Equilibrium Concepts so far and ahead A special case of Bayesian Games when no info. asym. Information So far = Complete/ Payoff functions are common knowledge Ahead = Incomplete/ Payoff functions are NOT common knowledge Time Static (SMG) Imperfect Info. Game. Nash Eq. Mixed Strat. NE. Bayesian Nash Eq. Dynamic (Seq.MG) Perfect Info. Game: Rollback Eq.=BIO. Sub-game Perfect Eq.(SPE) Perfect Bayesian Nash Eq. Imperfect Info Games We can use SPE for this game but the SPE may include non-credible threat. A special case of BNE when info asymmetry is 0

4.2.2 Uncertainty and Information Cases for Uncertainty on strategy or payoff Using mixed strategies by other players Ex) sports games, symmetric uncertainty  imperfect information: uncertainty on strategy Inherent in game itself by nature Ex) Organic food market, motives/types or payoffs are not common knowledge, asymmetric uncertainty  incomplete information: uncertainty on payoff Therefore, manipulation of information becomes an important strategy.

4.2.2 Uncertainty and Information Dealing with Risk for NE Ex) Assume you are player A. A has no dominant strategy but B has Left as a dominant strategy. Thus, B is expected to choose Left. Thus, A should choose Bottom  unique NE. But A does NOT observe what B chooses. Problem? NEXT. B Left Right A (You!) Top 9, 10 8, 9.9 Bottom 10, 10 -1000, 9.9

4.2.2 Uncertainty and Information Dealing with Risk for NE (cont’d) Lots of people will NOT choose Bottom if they were A. Why? B could think that there could be an estimation error or 10 and 9.9 could be switched by mistake. Thus, for these reasons, if B chooses Right? A receives the payoff of -1,000. This is a disaster for A. (낮은 확률이라도 죽을 가능성이 있으면…) 지금까지는 이러한 문제에 대응 못했음  베이지안 게임 도입 B Left Right A Top 9, 10 8, 9.9 Bottom 10, 10 -1000, 9.9

4.2.2 Uncertainty and Information Dealing with Risk for NE (cont’d) However, if there is any suspicion on payoffs or other player’s rationality, it should be reflected in the game. For example, A’s subjective belief tells that the probability for B’s payoff with Left(10) to be switched with B’s payoff with Right (9.9) (therefore, B chooses Right) is p. The probability that the payoffs are true (B chooses Left) as they are written is (1- p). 1- p p B Left Right A Top 9, 10 8, 9.9 Bottom 10, 10 -1000, 9.9 Prob. that the payoffs are true  Prob. that Left is dominant strategy is 1-p. Prob. that the payoffs are false  Prob. that Right is dominant strategy is p.

4.2.2 Uncertainty and Information Dealing with Risk for NE (cont’d) A does not know which is correct. Thus, A should try to maximize the expected payoff. The prob. for B to choose Left (dominant strategy) is (1-p) The prob. for B to choose Right (dominated strategy) is p. 1- p p B Left Right A Top 9, 10 8, 9.9 Bottom 10, 10 -1000, 9.9

4.2.2 Uncertainty and Information Dealing with Risk for NE (cont’d) Thus, A’s expected payoff for Top is 9*(1- p) if B chooses Left, 8* p if B chooses Right. Thus, A’s expected payoff for Top is 9*(1- p)+8* p. Likewise, A’s expected payoff for Bottom is 10*(1- p) * p. Thus, A should choose Top when the following condition is satisfied. 9(1- p )+8 p > 10(1- p)-1000 p  p >1/1009 1- p p B Left Right A Top 9, 10 8, 9.9 Bottom 10, 10 -1000, 9.9

4.2.2 Uncertainty and Information Dealing with Risk for NE (cont’d) That is, [A] should choose [Top] if A thinks that the probability that the payoffs are switched is higher than 1/1009(= 0.1%). In other words, even though the switching probability is very small (ex, 1%), choosing [Top] is more advantageous. That is, without considering the risk, choosing [Bottom] is best. However, after consideration of the risk, [Top] turns out to be a better choice. Choice of equilibrium strategy depends on probability  conditional equilibrium: the concept of BNE(Bayesian (Nash) Equilibrium)

4.2.2 Uncertainty and Information Dealing with Risk for NE (cont’d) This consideration, however, does not fall out of the basic rationality assumption of NE even though we assigned probabilities (subjective belief). Consider another example. Case 1) The prob. to earn or lose 1000KRW is 0.5 each. The expected payoff? 1000*0.5 – 1000*0.5 = 0. Case 2) The prob. to earn 10,010,000KRW is 50%. The prob. to lose 10,000,000KRW is 50%. The expected payoff is 10,010,000*0.5 – 10,000,000*0.5 =5000KRW Which case do you prefer? Case 1 or Case 2?

4.2.2 Uncertainty and Information Dealing with Risk for NE (cont’d) People should prefer case 2 b/c the expected payoff is greater in Case 2 (0<5000). But in practice, people prefer case 1  Most people are Risk-averse. Reasonable price for Lotto? The amount that people can forgo (to induce risk-averse people to buy lotto). How about 10,000,000KRW? (천만원짜리 로또는 당첨확률 높아도 안팔린다…) Lee, Byung-Huyun in ‘All-in.’: Bets by calculation. But executing calculated action is not for everyone. You need a lot of nerve (담이 커야…). Utility Risk Loving: Bet↑ U↑ at increasing rate Ex) squared U fn Risk Neutral: Bet↑ U↑ at same rate Ex) linear U fn Risk Averse: Bet↑ U↑, but at decreasing rate Ex) cube root U fn Bet in $

4.2.3 Concepts for Incomplete Information In many games, one or some player(s) have more precise information on what happened or will happen. Ex) In the Game of Chicken, each player knows better than the other players how courageous s/he her/himself is.(누구보다 자신이 자신을 더 잘 안다) That is, with more info., they take less risk or uncertainty. In other cases, strategies are unknown to other players. Even when strategies are common knowledge, the action by choice of strategy may not be observable. Ex) Principal-agent(주인-대리인) problem: You can’t observe all the detailed performance of your subordinates or agents. (부하/대리인이 일하는지 안하는지 상사는 잘 모른다… 바로 옆에서 봐도…)

4.2.3 Concepts for Incomplete Information (cont’d) Ex) In insurance, you can’t actually measure the exact (flood) damages of the insured, claiming compensation for a TV that was not actually owned by the insured. This way, manipulation of information on capability or preference of the players can affect the outcome of the game.  information manipulation becomes a strategy. Ex) 이전 게임에서, B가 바뀌었다고 생각하는 것처럼 signal 보낸 후에 Left 선택  보상 9.9  10 How? signaling and screening! For the players with more information; Conceal info or reveal misleading info (bluffing in poker) Reveal selected info truthfully (I don’t intend to fight even though the situation is volatile… US’s strategy to North Korean Nuclear Crisis)

4.2.3 Concepts for Incomplete Information (cont’d) For the players with less information; Elicit(끌어내다) info or filter truth from falsehood (Ex: 사랑한다면 우리집으로 이사와… capability of the new recruits, insurance company’s measure of the insured for the risk exposure.) Remain ignorant (Ex: Being unable to know your opponent’s strategic move can immunize you against his commitments and threats. Top-level politicians or managers often benefit from having “credible deniability.”  I have a low IQ and will go straight in the game of chicken.

4.2.3 Concepts for Incomplete Information (cont’d) How do you elicit or reveal info? Talk! But the problem is the credibility in talks. We will discuss the cases when the talk is credible. What if the talk is not credible? Actions speak louder than words. The less informed players should pay attention to what a better-informed player DOES(actions), not to what he says. The better informed players, knowing that the others will interpret actions this way, should in turn try to manipulate his actions for their information content. Available actions? NEXT.

4.2.3 Concepts for Incomplete Information (cont’d) Signaling : Actions that will induce other players to believe the info is good for you. Ex) North Korean’s missile launch to show they have nukes(good for them) Signal jamming : If others are likely to conclude that the information is bad for you, you may be able to stop them from making this inference by confusing them. Typically, a mixed strategy is used b/c random strategies make inference imprecise. Ex) The US said that it is not certain N. Koreans developed nukes. Screening : The strategy of making another player act so as to reveal his info. If others know more than you or take actions that you cannot directly observe, you can use this strategies to reduce your informational disadvantage. Ex) Move in, if you love me. Screening Device : specific methods used for screening. Ex) Which tire?

4.2.3 Concepts for Incomplete Information (cont’d) Incentive scheme: A strategy that attempts to influence an unobservable action of another player, by giving him some reward or penalty based on an observable outcome of that action (Ex: performance based pay system, Truth or Dare game; 진실게임) Truth or Dare: 인정 할 래, 시키는 대로 할 래?  인정하게 하기 위하여… One player starts the game by asking another player, "Truth or dare?" If the queried player answers, "truth," then the questioning player asks a question, usually embarrassing, of the queried player. If the queried player answers, "dare," then the questioning player asks the queried to do something, also usually embarrassing. After answering the question or doing the dare, the queried player asks "truth or dare?" of another player and the game proceeds as before. The game may be expanded upon by adding additional options to choose from, such as making the dare apply to both the questioner and the queried, or incorporating Spin the Bottle to select the next person to be asked; this results in many different "variants" of the game, limited only by the players' creativity.

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium. Recall that in the Battle of the Sexes there are two pure-strategy Nash equilibria. JP prefers the opera (payoff 2 for JP, 1 for JD), JD prefers K1(payoff 1 for JP, 2 for JD). If both go to the Opera, JP gets 2, JD gets 1. If both go to K1, JP gets 1, JD gets 2. Otherwise, payoffs are 0 for both. Payoff table? Below. JD Opera K1 JP 2, 1 0, 0 1, 2

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) If the payoffs are a common knowledge (complete –perfect-symmetric information), NE? (click) Mixed strategy equilibrium? JD Opera K1 JP 2, 1 0, 0 1, 2 q 1-q q-mix 2*q+0*(1-q), 1*q+0*(1-q) p 0*q+1*(1-q), 0*q+2*(1-q) 1-p 2*p+0*(1-p), 1*p+0*(1-p) 0*p+1*(1-p), 0*p+2*(1-p) p-mix

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) Both choose Opera Payoff = (2, 1) JP’s p-mix  equate the expected payoffs when JD chooses Opera or K1.  1*p+0*(1-p)=0*p+2*(1-p) => p=2/3(prob. to choose Opera) That is, the best strategy for JP is to choose Opera with prob. of 2/3 and to choose K1 with prob. of 1/3. JD’s q-mix : 2*q+0*(1-q)=0*q+1*(1-q) => q = 1/3 ; The best strategy for JD is to choose Opera with prob. of 1/3 and to choose K1 with prob. of 2/3. 1 JD’s BR 4/9 2/9 JD’s q-mix JP’s BR 1/3 2/9 1/9 1 2/3 JP’s p-mix Both Choose K1 Payoff = (1, 2)

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) Now, modify the game: JP and JD just started to be in relationship but both don’t know much about each other. In other words, if both go to Opera (altruistic choice by JD), JP may feel sorry for JD so that the payoff may be smaller than 2. Or JP is selfish so that the payoff may be higher than 2. This fact is only known to JP (JP’s private information). Likewise, if both go to K1, JD’s payoff can be higher or lower than 2. Let’s assume their payoffs are higher than 2 for now (or assume both are selfish).

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) Thus, the payoff table is modified as follows. tJP : increase in JP’s payoff (known only to JP) tJD : increase in JD’s payoff (known only to JD). That is, the information on tJP and tJD are incomplete-asymmetric. For example, tJP and tJD are numbers between [0, 1], if they are selfish. JD Opera K1 JP 2+tJP, 1 0, 0 1, 2+tJD

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) For example, if tJP =0.5 and tJD =0.1, the payoff table is as follows. Also assume that JP will choose Opera, if the value of tJP is greater than a value, say b, that is between [0, 1]. *b=JP의 이기심 발동시키는 추가보상 임계치 JD also chooses K1, if the value of tJD is greater than a value, say p, that is between [0, 1]. *p=JD의 이기심 발동시키는 추가보상 임계치 Example JD Opera K1 JP 2+0.5, 1 0, 0 1, 2+0.1

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) As in the figure below, JP will choose Opera (act selfish, bearing the shame) if the increase in payoff is greater than b. JP will choose K1 (patronizing JD) if the increase is less than b. Thus, the prob that JP chooses Opera is (1-b)/1 and the prob. that JP chooses K1 is b/1. For JP Choose K1 Choose Opera tJP < b b < tJP tJP b (1-b) b 1

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) The same for JD. JD will choose K1(act selfish bearing the shame) if the increase in payoff is greater than p. JD will choose Opera if the increase is less than p. Thus, the prob that JD chooses Opera is p/1 and the prob that JD chooses K1 is (1-p)/1. For JD Choose Opera Choose K1 tJD < p p < tJD tJD p (1-p) 1 p

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) But the change in payoff can be bigger or smaller than 1. Thus, let x be the change in payoff in general. Prob for JP to choose Opera: (x - b)/ x Prob for JP to choose K1: b/x Prob for JD to choose K1: (x - p)/x Prob for JD to choose Opera: p/x Choose K1 Choose Opera Choose Opera Choose K1 tJP < b b < tJP tJD < p p < tJD tJP tJD x x b p

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) JP’s expected payoff in this case? The expected payoff for JP to choose Opera The expected payoff for JP to choose K1 Prob for JD to choose Opera Prob for JD to choose K1 JD Opera K1 JP 2+tJP, 1 0, 0 1, 2+tJD Prob for JP to choose Opera (x - b)/x b/x p/x (x - p)/x Prob for JD to choose Opera

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) Thus, if JP should choose Opera. The above can be rearranged into Why b=x/p-3 ? b/c it is assumed than JP chooses Opera when tJP is greater than b. 오페라 선택시 JP의 기대보상 K1 선택시 JP의 기대보상

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) JD’s expected payoffs? The expected payoff for JD to choose K1 The expected payoff for JD to choose Opera JD Opera K1 JP 2+tJP, 1 0, 0 1, 2+tJD (x - b)/x b/x p/x (x - p)/x

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) JD should choose K1 if Rearranging the above, we have, K1 선택시 JD의 기대보상 오페라 선택시 JD의 기대보상

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) Rearranging again,

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) * p, b are positive ( p, b > 0)

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) That is, JP will choose Opera if his additional payoff (tJP) is greater than b. JD will choose K1 if her additional payoff (tJD) is greater than p. (We assumes this) However, even though they know their own t, the choice of strategy depends on x. x is the measure of info asymmetry.(정보비대칭의 크기) : JP chooses Opera. : JD chooses K1.

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) Bayesian Nash Equilibrium (BNE) What if x is 1? : JP chooses Opera. 둘 다 덜 이기적  추가보상(b, p) 이 꽤 커야 이기적으로 행동 Choose K1 Choose Opera : JD chooses K1. 0 b= What if x is 0.001(decreased info asymmetry or incompleteness)? Bayesian Nash Equilibrium (BNE) : JP chooses Opera. 둘 다 매우 이기적  조금 만 보상이 늘어도 자기가 가고 싶은 대로 각자 가자고 함 Choose K1 Choose Opera 0 b= : JD chooses K1.

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) If x converges to 0(smaller and smaller asymmetry), BNE and Mixed Strategy Nash Equilibrium converges. (x - b)/ x p=2/3

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) That is, Mixed Strategy Nash Eq.(MSNE) is equivalent to Bayesian Nash Eq.(BNE) with information asymmetry/incompleteness of 0 (x0). The relationship is presented in the figure below. BNE MSNE =BNE w/ Info Asymmetry 0

4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) We also learn that NE is a subset of MSNE. Recall that MSNE can represent pure strategies with probability O or 1 for NE. That is, NE is a special case of MSNE with probabilities are assigned only for 0 or 1. Thus, the following figure can be drawn. MSNE NE = MSNE with prob = 0 or 1

Backward Induction Outcome
4.2 Static Games under Incomplete Information 4.2.4 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium (cont’d) [End of the Chapter] Finally, integrating the previous figures, we have the following relationship among equilibrium concepts. BNE MSNE NE Subgame Perfect NE Rollback Eq. or Backward Induction Outcome

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Another example of Incomplete info game: a modification of Cournot(1838) Game. Rules of the Normal Form game Player : 2 companies(1 and 2) Strategies : quantity of production Payoff : profit Price is determined by demand fucntion: P(Q) = a - Q Total quantity produced is summation of two companies’ quantity produced: Q = q1+ q2 Cost: Ci(qi) = ciqi or C1(q1) = c1q1

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP 1 maximizes its profit. The condition for max. profit is  Similarly, 2’s condition is 

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP NE is the combination of quantities that maximize each company’s profit simultaneously, (q1* and q2*). Solving the 2 previous equations, Similarly, In general,

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Under symmetric info (1 and 2 know other company’s cost structure. C1 and C2 are known to each other), the outcome of the original Cournot game is as follows and it is the NE. My quantity (qi) decreases if my unit cost (ci) increases AND increases if other company’s unit cost (cj) increases.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Now the original Cournot Game is modified, introducing info asymmetry. Now, 1 know its own cost structure but does not know 2’s cost structure (high or low). But 2 not only knows its own cost structure but also knows 1’s cost structure. info asymmetry. For example, 2 is a new entrant in the market and 1 does not know 2’s cost structure. Or 1 may think that 2 developed a new production technology to lower the production cost, but 1 does not know for sure.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Like before, price is determined by demand: P(Q) = a – Q Q = q1+ q2 Costs are a little different, though. That is, 1’s cost structure: C1(q1) = cq1 2’s cost structure: High: C2(q2) = cHq2, prob = θ Low: C2(q2) = cLq2, prob =1-θ where, cH > cL *1 knows the probabilities only.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Under the circumstance, 1 is not sure about 2’s cost structure TYPE(High cost structure type or Low cost structure type) but knows the probabilities of the types.  1 will try to max its expected profit. Thus, 1 maxmizes; Expected Profit=Prob of High*Profit of High+Prob of Low*Profit of Low In an equation, (1)

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP 2 know its own type for sure. Thus, profit max problem for 2 when its cost is High, If the cost is Low, (2) (3)

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Differentiating (1), 1’s profit max quantity (q1) is, *Compare this with that of symmetric info:

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP 2’s profit maximizing quantity is decided by differentiating (2) when 2 is High cost type. * Compare this with that of symmetric info:

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP 2’s profit maximizing quantity is decided by differentiating (3) when 2 is Low cost type. The quantity that satisfies the above 3 equations is the quantity at NE. Thus, solving the equations for quantities gets you the best response functions. Replacing 1’s BR function with 2’s BR function, the following is acquired.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP That is, the quantity when 1 maximizes its profit is,

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Next is the quantity when 2 maximizes its profit for the case of High cost structure.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Next is the quantity when 2 maximizes its profit for the case of Low cost structure.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP The below table shows the comparison with the NE under symmetric info. Complete info or symmetric info Incomplete info or asymmetric info Or in general, Case of High cost, Case of Low cost,

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Implications If 1 knows that 2 is High cost type for sure (θ =1, or no info asymmetry), That is, BNE = NE with symmetric info.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP If 1 knows that 2 is Low cost type for sure (θ =0, or no info asymmetry), Again, BNE = NE Thus, BNE generalizes NE under symmetric info. The problem arises when 1 is NOT sure about the type of 2 (i.e. 0 < θ <1). In this case, what is the implication?

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP Compared to the quantity on , when 2 knows the cost structure (type) of 1 for sure, 2 produces more if 1’s type is High, 2 produces less if 1’s type is Low, Slide 39

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP That is, 2 is better-off from info asymmetry if he is High type, and worse-off if he is Low type. Thus, 2 should reveal info on its cost advantage to 1 when 2 is High type, and should employ signal jamming to confuse 1 when 2 is Low type. Also, 2 knows its own type for sure. In this case, Thus, 2 knows it can increase production when the type is Low cost.

4.2.5 Games with Incomplete/Asymmetric Information: Bayesian Nash Equilibrium : Example SKIP But because 1 is not sure, compared to the case when 1 knows 2’s type, 2 produces; more (than the case that 1 knows 2’s type for sure) when it is High type, less (than the case that 1 knows 2’s type for sure) when it is Low type. This approach for SMG under asymmetric info can be applied to auction, because, in auction, participants are not sure about other players’ payoff.

4.2 Static Games under Incomplete Information Information

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