Question

Tom, ryan and Harry are the three City Council members for Smallsville. They would all like a

raise but know that voting for one will ensure their defeat in the next election. If they vote

against it, their re-election is certain. Each earns five thousand dollars per year and has two years

left on his term. A raise would increase the pay by one thousand dollars per year. The next term

is for three years, following which none of them could run again. The raise will be passed if at

least two of the Council members vote for it. The vote will be taken by the roll-call method

whereby each player in turn must announce his vote (yes or no) and the other players are able

to listen. Assuming that Tom votes first, then ryan, and finally Harry, draw the game tree and

solve the game by backward induction. Assume that these politicians are only in it for the

money. Is there any first mover advantage or disadvantage?

Answer 1

IF THEY VOTE YES THEN THEY ARE NOT RE-ELECTED SO THE NUMBER OF YEARS THEY ARE IN POSITION IS 3.

HOWEVER, IF THEY VOTE AGAINST i.e. NO THEN THEY ARE RE-ELECTED AND THE NUMBER OF YEARS THEY ARE IN POSITION IS 6.

FOR EVERY YEAR THEY ARE IN POSITION THEY GET 5K $.  

THE RAISE IS PASSED IF AT LEAST 2 MEMBERS VOTE YES . WHEN THE RAISE IS PASSED, MEMBERS ARE GIVEN ADDITIONAL 1K$ IN EACH TERM.

FIND ATTACHED THE GAME TREE.

BY BACKWARD INDUCTION, WE SOLVE FOR THE GAME AND THE SOLUTION IS CALLED SPNE i.e. SUBGAME PERFECT NASH EQUILIBRIUM.

HERE THE SPNE TURNS OUT TO BE WHEN ALL MEMBERS VOTE AGAINST(NO) WITH PAYOFFS (30,30,30).

PATH MARKED WITH ORANGE IS THE EQUILIBRIUM PATH FOLLOWED. SO THE EQUILIBRIUM STRATEGY FOR {TOM, RYAN, HARRY} = {NO , NO , NO}

THERE IS NO FIRST MOVER ADVANTAGE / DISADVANTAGE SINCE IN EQUILIBRIUM ALL RECEIVES EQUAL PAYOFFS (=30)