Question

Consider a game between a tax collector (player 1) and a tax payer (player 2).  Player2 has an income of 200 and may either report his income truthfully or lie.  If he reports truthfully, he pays 100 to player 1 and keeps the rest.  If player 2 lies and player 1 does not audit, then player 2 keeps all his income.  If player 2 lies and player 1 audits then player 2 gives all his income to player 1. The cost to player 1 of conducting an audit is 20.  Suppose that both parties move simultaneously (i.e.  player 1 must decide whether to audit before he knows player 2’s reported income).  Find the mixed-strategy Nash equilibrium for this game and the equilibrium payoffs to each player.  Explain in your own words the meaning of these results.

Answer 1

Payoff table: Player II Strategy Таx рayer Player I Strategy Tax collector Probability Truthful Lie Audit 100 X2=200 No audit 1-P X3:=100 Probability 1-P2 Р2

Maximin Decision Criterion to the strategies for Player I Minimum value of "Audit" strategy: Мa3тin (r,, г.) — 100 Minimum value for strategy "No Audit" MNA-тin ('3, т) 3D0 Оp3max (Mд, MNA) 3 100 Maximum of these two minimum values Ор, — 100 optimal for Player I with strategy: Payoff Table with Maximin Criterion of Player I Player II Strategy Tax payer Player I.Strategy Tax collector Probability Truthful Lie Audit C7=100 200 P1

No audit 1-P1 3=100 0=x Probability 1-P2 Р2 Maximum of these two minimum values Оp, 3D max (Mд, MNA) — 100 Minimax Decision Criterion to the strategies of Player II: Maximum value of "Truthful" strategy: Мтi-max (х,, 3) — 100 Maximum value for strategy "Lie" : Mi3max (х,, т) — 200 Minimum of these two maximum values Ор,:—тin (Mт,Mм) — 100 optimal for Player II with strategy: Оp, 3 100

Payoff Table with Minimax Criterion of Player II Player II Strategy Tax payer Player I Strategy Tax collector Probability Truthful Lie Audit X1100 200 P1 No audit 1-P 3=100 Probability 1-P2 Р2 Ор,:3D тin (Mт, M) — 100 Minimum of these two maximum values

Payoff Table with combined strategies of Player I and II Player II Strategy Tax payer Player I Strategy Tax collector Probability Truthful Lie Audit Г2:-200 E100 No audit 1-P X3:= 100 0='x Probability 1-P2 Ор. - тin (t'3,) —0 Minimum of these two maximum values + From Payoff Table with combined strategies of Player I and II, that the strategies selected by the tax payer and tax collector do not result in an equilibrium point. and so, is not a pure strategy game.

Payoff table: Player II Strategy Tax payer Player I Strategy Tax collector Probability Truthful Lie Audit 100 X2 200 P1 1-P No audit X3100 Probability 1-P2 P2

Expected Gain for player I: Probability of choosing strategy "Audit": Probability of choosing strategy "No audit": 1-Р1 If Player I selects "audit", and player selects "Truthful", then expected gain for player I 100-р, + 100- (1—р,) %3D 100 Next, Player I assumes that Player II will select strategy Lie 200-р, +0.(1—р.) %3D 200-р,

If Player I is same if player II selects strategy Truthful or Lie,we equate the expected gain from each of these strategies: 100 =0.5 200 100 200.p gain for player I P1 So, We can find that the player I is planned to use Truthful or Lie for 50% of the time he reports his income tax. EG (PIA) 3D P1 (2,) + (1 — р) (тз) %— 100 EG (PINA)=p1 (x2) + (1-p,)- (x,) =100

Repeat this process for Player II to find out his mixed strategy The expected loss for Player II, for being Truthful 100-р2 + 200- (1— рә) %3D 200— 100-р, The expected loss for Player II, for being a liar 100- Р, +0.(1— Рә) — 100-Р> 200 1 200 200 100 p2= 100-p2 Equate these two losses P2 EL (PI) :- P2: (г,) + (1— р.) (т-ә) %3 100 EL (PI) :3D P2: () + (1 — рә) - (т,) — 100

There is a simultaneous gain of 100 for player I and loss of 100 for player II, so the mixed strategy of the two players has come to an equilibrium point.