Lecture 6 Dynamic games with imperfect information

Lecture 6 Dynamic games with imperfect information

Backward Induction in dynamic games of imperfect information We start at the end of the trees first find the Nash equilibrium (NE) of the last subgame then taking this NE as given, find the NE in the second last subgame continue working backwards If in each subgame there is only one NE, this procedure leads to a Unique Subgame Perfect Nash equilibrium

Example: two stage game of imperfect information Stage : Players and move simultaneously taking, respectively, actions a A and a A Stage : Players and 4 observe (a, a ), then move simultaneously taking, respectively, actions a A and a 4 A 4 Payoffs: u i (a, a, a, a 4 ) for i =,,, 4 Solution: We solve the simultaneous - move game between players and 4 in the second stage: Players and anticipate the behaviour of players and 4

4 Player Player Player Player 4 T B T B T B T B T B L R L R L R L R L R L R L R L R L R L R 4 4 4 4

Player Player L R T G G B G G4 G Player 4 L R Player T,,,,,, B,,, 4,4,, G Player 4 L R Player T,,,,,, B,,,,,, 5 G Player 4 L R T,,,,,, Player B 4,4,,,,, G4 Player 4 L R T,,,,,, Player B,,,,,,

Player Player L R T,,,,,, B,,,,,, G Player 4 L R Player T,,,,,, B,,, 4,4,, G Player 4 L R Player T,,,,,, B,,,,,, 6 G Player 4 L R T,,,,,, Player B 4,4,,,,, G4 Player 4 L R T,,,,,, Player B,,,,,,

Backward induction outcome: (B, L, T, L) Subgame perfect Nash equilibrium (B, L, (T, T, T, B), (R, R, L, L)) 7

Example Challenger s strategies: {(Out Ready), (Out Unready) (In ready), (In Unready)} Incumbent strategies: Acquiesce, Fight Challenger In Challenger Out Ready Unready Incumbent Acquiesce Fight Acquiesce Fight 4 4 8

Challenger Incumbent Acquiesce Fight Out Ready, 4, 4 Out Unready, 4, 4 In Ready,, In Unready 4,, Three Nash equilibria:. (Out Ready, Fight);. (Out Unready, Fight). (In unready, Acquiesce) 9

Consider the subgame starting in the decision node after Challenger s choice In Incumbent Challenger Acquiesce Fight Ready,, Unready 4,, An unique Nash equilibrium: Unready, Acquiesce Then, only (In unready, Acquiesce) is subgame perfect NE

Example Incumbent Acquiesce Fight Challenger Ready,, Unready 4,, Out, 4, 4 Two Nash equilibria: (Out, Fight) (Unready, Acquiesce) Challenger Both are SPNE Ready Unready Incumbent Acquiesce Fight Acquiesce Fight Out 4 4

Applications with imperfect information

Bank Runs Two investors, one bank Each investor has deposited D with a bank The bank has invested D in a long term project If the bank liquidates the investment before the end, it will get back r, where D/ < r < D Otherwise the bank will get R, where R > D

Investors can make withdrawals from the bank at: Date, before the end of the investment Date, after the end of the investment It is enough that one investor makes withdrawal at date to force the bank to liquidate the investment 4

Payoffs: Both investors make withdrawals at date : each one receives r. Only one investor makes withdrawal at date : he receives D, the other receives r D. Neither investor makes withdrawal at date : Both investors will take a withdrawal decision at date Both investors make withdrawals at date : each receive R Only one investor makes withdrawal at date : he receives R - D, the other receives D. Neither investor makes withdrawal at date : The Bank returns R to each investor 5

Investor Date Investor Withdraw No withdraw Withdraw r, r D, r - D No withdraw r D, D Next stage Investor Date Investor Withdraw No withdraw Withdraw R, R R D, D No withdraw D, R D R, R 6

We solve the game in date Investor Date Investor Withdraw No withdraw Withdraw R, R R D, D No withdraw D, R D R, R In date s game there is only one Nash equilibrium: {(Withdraw), (Withdraw)} where each Investor gets R 7

In date the two investors anticipate that in the case neither investor makes withdrawal at date, the game goes in the second stage (date ) and that in date two the outcome will be (the NE): {(Withdraw), (Withdraw)} where each Investor gets R. Then the game in date can be written as: Investor Date Investor Withdraw No withdraw Withdraw r, r D, r - D No withdraw r D, D R, R 8

We solve the game in date : Investor Date Investor Withdraw No withdraw Withdraw r, r D, r - D No withdraw r D, D R, R There are two Nash equilibria in date game: {(Withdraw), (Withdraw)} {(No withdraw), (No Withdraw)} 9

Game in Date, one NE: {(Withdraw), (Withdraw)} Game in Date (reduced), two NE:. {(Withdraw), (Withdraw)}. {(No withdraw), (No Withdraw)} Whole game: Two Backward Induction Outcomes (BIO): ) {(Withdraw), (Withdraw)} in date ) {(No withdraw), (No Withdraw)} in date, {(Withdraw), (Withdraw)} in date Two subgame perfect NE (SPNE): ) {(Withdraw, Withdraw), (Withdraw, Withdraw)} ) {(No withdraw, Withdraw), (No Withdraw, Withdraw)} Note SPNE ) supports BIO ), SPNE ) supports BIO )

Extensive form representation Withdraw Investor Investor No Withdraw r r W NW D r-d W r-d D NW Investor W NW Investor W NW W NW R R R D D D R D R R Investor : information sets Investor : information sets Investor s strategies: {(W, W), (W, NW), (NW, W), (NW, NW)} Investor s strategies: {(W, W), (W, NW), (NW, W), (NW, NW)}

Backward Induction Withdraw Investor Investor No Withdraw r r W NW D r-d W r-d D NW Investor W NW Investor W NW W NW R R R D D D R D R R Investor : information sets Investor : information sets Investor s strategies: {(W, W), (W, NW), (NW, W), (NW, NW)} Investor s strategies: {(W, W), (W, NW), (NW, W), (NW, NW)}

Reduced game Withdraw Investor Investor No Withdraw W NW W NW r r Withdraw D r-d r-d D Investor Investor R R No Withdraw W NW W NW r r D r-d r-d D R R Withdraw Investor Investor No Withdraw W NW W NW r r D r-d r-d D R R

Tariffs and Imperfect international competition Two identical countries denoted by i =,. One homogeneous good is produced in each country by a firm, firm i in country i A share h i of this product is sold in the home market and a share e i is exported in the other country Governments choose tariffs, i.e. a tax on the import. Government of country i chooses tariff t i. 4

In country i the market clearing price is: P i (Q i ) = a Q i where Q i = hi + ej Firms have constant marginal cost, c, and no fixed cost Firm s payoff (profits): i = [a hi e j ]hi + [a h j e i ]e i c[hi + e i ] t j e i Government s payoff (consumer welfare + home firm s profit + tariff revenue) W i =.5 Qi + i + t i e j 5

Timing. Governments simultaneously choose tariffs (t, t ). Firms observe (t, t ) and simultaneously choose quantities h, e, (h, e ). Backward induction solution. We suppose that governments have chosen tariffs (t, t ) and we find the optimal behaviour of firms as function of (t, t ).. We assume that governments correctly predict the optimal behaviour of firms for each possible combination of (t, t ) and we find the optimal tariff rates. 6

We suppose that governments have chosen tariffs (t, t ) and we find the optimal behaviour of firms as function of (t, t ). max h,e where = [a h e ]h + [a h e ]e c[h + e ] t e Firm s FOCs: [a h e ] c = [a h e ] c t = h = (a e c) / e = (a h c t ) / 7

For Firm : max h,e where = [a h e ]h + [a h e ]e c[h + e ] t e Firm s FOCs: [a h e ] c = [a h e ] c t = h = (a e c)/ e = (a h c t )/ 8

We have to solve a system of 4 equations in 4 unknowns:. h = (a e c) /. e = (a h c t ) /. h = (a e c) / 4. e = (a h c t ) / Solutions:. h *= (a c + t ) /. e *= (a c t ) /. h *= (a c + t ) / 4. e *= (a c t ) / 9

We assume that governments correctly predict the optimal behaviour of firms for each possible combination of (t, t ) and we find the optimal tariff rates. The problem of country s government is: max W =.5 (Q ) + + t e t where Q = h + e = (a c + t ) + (a c t ) = (a c t ) = [a h e ]h + [a h e ]e c[h + e ] t e Using algebra W = ((a c) t ) 8 + (a c+t ) 9 + (a c t ) 9 + t (a c t )

Similarly we can write the problem of country s government We compute the governments FOCs and we find: t * = (a c)/ t * = (a c)/ Then Firm will produce: h * = 4(a c)/ 9 e * = (a c)/ 9 Firm will produce: h * = 4(a c)/ 9 e * = (a c)/ 9

Backward Induction outcome t * = (a c)/ t * = (a c)/ h * = 4(a c)/ 9 e * = (a c)/ 9 h * = 4(a c)/ 9 e * = (a c)/ 9

Subgame Perfect Nash Equilibrium (SPNE): Note: One info set for governments infinite number of info set for firms, i.e. each possible combination of t t t * = (a c)/ t * = (a c)/ h *= (a c + t ) / e *= (a c t ) / h *= (a c + t ) / e *= (a c t ) /