EconS Advanced Microeconomics II Handout on Bayesian Nash Equilibrium

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1 EconS Avance icroeconomic II Hanout on Bayeian Nah Equilibrium 1. WG 8.E.1 Conier the following trategic ituation. Two oppoe armie are poie to eize an ilan. Each army general can chooe either "attack" or "not attack." In aition, each army i either "trong" or "weak" with equal probability (the raw for each army are inepenent), an an army type i known only to it general. Payo are a follow: The ilan i worth if capture. An army can capture the ilan either by attacking when it opponent oe not or by attacking when it rival oe if it i trong an it rival i weak. If two armie of equal trength both attack, neither capture the ilan. An army alo ha a "cot" of ghting, which i if it i trong an w if it i weak, where < w: There i no cot of attacking if it rival oe not. Ientify all pure trategy Bayeian Nah equilibria of thi game. Anwer: There are four pure trategie contingent on the type of player: AA : Attack if either weak or trong type, AN : Attack if trong an Not attack if weak, NA : Not attack if trong an Attack if weak, NN : Never attack. We can etermine the expecte payo for each player by imple calculation. For example, the expecte payo for player 1 playing the trategy AA given that player alo play the trategy AA i EU 1 (AAjAA) = Player 1 i Strong z} { 0:5 3 Player i Weak z } { z } { 0:5 ( ) + 0:5 ( ) 5 + 0:5 [0:5 ( w) + 0:5 ( w)] Player i Strong = + w The remaining expecte payo for each pair of trategie can be eaily compute an are given in gure 1: 1

2 Player AA AN NA NN AA ; ; 3 ; w ; 0 Player 1 AN ; ; ; w ; 0 NA w ; 3 w; w; w ; 0 NN 0; 0; 0; 0; 0 Figure 1: Normal Form Repreentation Any NE of thi normal form game i a Bayeian NE of the original game. [Cae 1] > w > ; an w > > From the above payo we can ee that (AA; AN) an (AN; AA) are both pure trategy Bayeian Nah equilibriua. [Cae ] > w > ; an < From the above payo we can ee (AA; NN) an (NN; AA) are both pure trategy Bayeian Nah equilibria. [Cae 3] w > > ; an < From the above payo we can ee that (AN; AN) ; (AA; NN) an (NN; AA) are pure trategy Bayeian Nah equilibria. [Cae ] w > > ; an > From the above payo we can ee that (AA; AN) ; (AN; AA) an (AN; AN) are pure trategy Bayeian Nah equilibria.. WG 8.E.3 Conier the linear Cournot moel ecribe in Ex 8.B.5; two rm 1 an ; imultaneouly chooe the quantitie they will ell on the market, q 1 an q : The price each receive for each unit given thee quantitie i P (q 1 ; q ) = a b (q 1 + q ) : Their cot are c per unit ol. Now, however, uppoe that each rm ha probability of having unit cot of c L an (1 ) of having unit cot of c H ; where c H > c L : Solve for the Bayeian Nah equilibrium.

3 Anwer: A rm of type i = H or L will maximize it expecte pro t, taken a given that the other rm will upply q H or q L epening whether thi rm i of type H or L: A type i fh; Lg rm 1 will maximize: ax(1 ) [(a b (q 1 qi 1 i + qh ) c i)qi 1 ] + [(a b (qi 1 + ql ) c i)qi 1 ] The FOC yiel: (1 )(a b (q 1 i + q H ) c i) + (a b (q 1 i + q L ) c i) = 0 In a ymmetric Bayeian Nah equilibrium: q 1 H = q H = q H an q 1 L = q L = q L Plugging thi into the F.O.C we get the following two equation: Therefore, we obtain that (1 ) [a 3bq H c H ] + [a b (q H + q L ) c H ] = 0 (1 ) [a b (q H + q L ) c L ] + [a 3bq L c L ] = 0 3. Auction Theory q H = a c H + (c L c H ) 1 ; 3b q L = a c L + 1 (c H c L ) 1 3b : Conier a rt-price eale-bi auction in which bier imultaneouly ubmit bi an the object goe to the highet bier at a price equal to hi/her bi. Suppoe that there are N bier, whoe type are i are rawn from the cumulative itribution F (:) an receive utility from the object from function u( i b), where b i the amount of the bi pai by player i.. a) Solve for 0 (), the erivative of the biing function of each iniviual. 3

4 Anwer: We tart by etting up the utility maximization problem for player i, max b0 E i [v i (b; i ( i ); i )j i ] = [F ( 1 (b))] N 1 u( i b) Taking rt-orer conition, (N 1)[F ( 1 (b))] N f( 1 1 (b) u( i b) [F ( 1 (b))] N 1 u 0 ( i b) = for () to be an optimal biing function, it houl be optimal for the bier not to preten to have a valuation i erent from hi real one, i. Hence () = b i the optimal olution for the above rt-orer conition, an we have that 1 (b) =, implying Simplifying, an olving for 0 () yiel (N 1)[F ()] N f()u( ()) 0 () (N 1)f()u( ()) 0 () [F ()] N 1 u 0 ( ()) = 0 = F ()u 0 ( ()) 0 () = u( u 0 ( ()) ()) f() (N 1) F () b) Aume that conumer are rik neutral, i.e. u(x) = x. Derive the biing function (). Anwer: Uing what we erive in part (a), we have rearranging ome term, 0 () = ( ()) f() (N 1) F () f()()(n 1) + F () 0 () = f()(n 1) we can multiply both ie by [F ()] N to obtain [F ()] N f()()(n 1) + [F ()] N 1 0 () = [F ()] N f()(n 1) Note that the left ie i now jut ([F (x)] N ([F (x)]n 1 1 ()) ()). Subtituting, = [F ()] N f()(n 1)

5 an integrating both ie let [F (x)] N 1 () = [F (x)] N xf(x)(n 1)x [F (x)] N 1 () = [F (x)] N 1 h(x) = x g 0 (x) = [F (x)] N f(x)(n 1)x h 0 (x) = x g(x) = [F (x)] N 1 an applying integration by part, we have [F (x)] N 1 x an olving for () yiel the biing function () = 1 [F (x)] N 1 [F (x)] N 1 x c) Now aume that conumer are rik avere, i.e. u(x) = x where 0 < < 1. Derive the biing function (). Anwer: Uing what we erive in part (a), we have rearranging ome term, 0 () = 1 ( f() ()) (N 1) F () f()() 1 (N 1) + F ()0 () = f() 1 (N 1) we can multiply both ie by [F ()] 1 (N 1) 1 to obtain [F ()] 1 (N 1) 1 f()() 1 (N 1) + [F ()] 1 (N 1) 0 () = [F ()] 1 (N 1) 1 f() 1 (N 1) Note that the left ie i now jut ([F (x)] 1 (N 1) ()). Subtituting, ([F (x)] 1 (N an integrating both ie 1) ()) = [F ()] 1 (N 1) 1 f() 1 (N 1) [F (x)] 1 (N 1) () = [F (x)] 1 (N 1) 1 xf(x) 1 (N 1)x 5

6 let h(x) = x g 0 (x) = [F (x)] 1 (N 1) 1 f(x) 1 (N 1)x h 0 (x) = x g(x) = [F (x)] 1 (N 1) an applying integration by part, we have [F (x)] 1 (N 1) () = [F (x)] 1 (N 1) [F (x)] 1 (N 1) x an olving for () yiel the biing function () = 1 [F (x)] 1 (N 1) [F (x)] 1 (N 1) x 6