Question 1 (25 marks)
The Higher Salaries Commission (HSC) determines salaries for politicians and other high ranking public sector officials in NZ. Imagine the HSC has 3 members, two from the North island - A from Auckland, W from Wellington - and one from the South Island - C from Canterbury. A, W and C are all politicians themselves. Generally each of them prefers a higher salary to a lower salary other things equal. But in the world of politics "other things" aren't usually equal. Each HSC member well knows that he/she may face a political backlash when the press or television media reports the committee's voting behaviour. If an HSC member is "seen" by her electorate to be voting to increase her own salary her political career will be put at risk.
HSC members meet together around a small table, face to face. A minute keeper is present to keep detailed minutes of the meeting, including a record of how each member of the committee votes. These minutes are traditionally made available publicly by posting them on committee's Internet web page after a meeting. Suppose that after discussing the pros and cons of salary increases, the recent history of salary changes, inflation, and other related issues the HSC comes down to two options: either a 5% salary increase backdated to Jan 1, 2001, or no salary increase at all this year. The actual method for making a decision by the committee is roll-call voting. The chairperson moves a formal motion "That the HSC recommend a salary increase" and each member can vote F="For" or A="Against" at his/her turn. Traditionally voting proceeds in order from the "top" to the "bottom" of NZ: first A then W then C. Later voters can see how earlier voters voted before casting their votes. A simple majority is required for the motion to pass - otherwise it fails.
Suppose the preference rankings of each HSC member are as listed in the following table (higher numbers indicate higher ranked outcomes):
| The salary raise motion passes but one's own vote is A=Against | 4 |
| The salary raise motion fails and one's own vote is A=Against | 3 |
| The salary raise motion passes but one's own vote is F=For | 2 |
| The salary raise motion fails and one's own vote is F=For | 1 |
1a How many strategies does each player have?
1b Construct and label a game tree for this problem and find the rollback equilibrium using both pruning & arrows.
1c Explain how you would analyse this voting situation if the committee's voting procedure changed from roll call voting to secret ballot voting. Here, each member records her vote on a piece of paper (a ballot) in a manner not observable to others, places it in a sealed envelope, and passes his/her envelope to the minute keeper. The minute keeper leaves the room and under the watchful eye of an independent scrutiniser counts the votes. She reports back only whether the motion passed or failed, and only the result is publicly reported - i.e. no detail is provided on the extent of the winning margin or on how individual committee members voted.
1a: (1+2+2=5 points) A has 2 moves and because she moves first, 2 strategies; W moves second and can observe A's moves, so has 2 moves for each of A's 2 moves, so 4 strategies altogether; C moves third and can observe what both A and W have chosen - there are 4 such possibilities (FF,FA,AF,AA where the first letter indicates A's move and the second letter W's move) so 16 strategies for C. Note: I asked only for the number of strategies, not a detailed specification) - which can get tedious. It may also be easier to "count" these strategies after constructing the game tree.
1b (12 points) see the attached game tree (which by and large was well done for the entire class) ; all your nodes and branches must be clearly labelled . Rollback indicates
- the equilibrium combination of strategies Against for A, always Against for W (no matter what A does), and always Against for C (no matter what A or W do),
- an equilibrium path of play where all vote against (linked arrows in the tree), and
- equilibrium payoffs are (3,3,3), third highest for each politician. A rollback equilibrium, like any equilibrium, is specified by the strategy choices of each agent -not the path of play nor the payoff distribution. I subtracted 3 points for NOT indicating what these strategies were but getting everything else correct (quite common: many of you will have something like 12-3=9 recorded on your sheet with a "see ans guide"). Note that the equilibrium strategies , as strategies, MUST specify what players will do in all the situations they could find themselves in).
- c (8 points, divided 4 and 4 according to the following ways of analysing the game) there are several ways to approach this question. Remember the structure of a game (players, what they can do, what they know when it is their turn to make a decision, and what their payoffs are). W and C no longer get to observe the others' votes as in roll call voting. Hence we want to analyse this as a simultaneous game. But what are the payoffs? One could assume that the only thing that has changed is the information conditions, leading to the following 3 person simultaneous game. I have marked payoffs "xyz" are x for A, y for W, and z for C following the usual conventions.
| C chooses F | C chooses A | |||||
|---|---|---|---|---|---|---|
| W chooses | W chooses | |||||
| A chooses | F | A | F | A | ||
| F | 222 | 242 | F | 224 | 133 | |
| A | 424 | 331 | A | 313 | 333 | |
Find each player's best responses to startegies of the other players (I have marked the cells with "a" for A's best response, "w" for W, "c" for C; if you don't understand how these best responses are calculated please come and see me). The only mutually consistent best response is when all vote A=Against (as indicated in the bottom right cell).
| C chooses F | C chooses A | |||||
|---|---|---|---|---|---|---|
| W chooses | W chooses | |||||
| A chooses | F | A | F | A | ||
| F | 222 | w 242 | F | c224 | wc133 | |
| A | a 424 | aw 331 | A | ac313 | awc333 | |
However, why assume the payoffs remain unchanged? After all, the description of the game indicated that the politicians preferred higher salaries but they just don't want to be seen voting for higher salaries themselves. The minute keeper only records the aggregate result. Clearly if it passes unanimously, from strategy choices FFF, then the electorate will know who voted which way, so A prefers AFF to FFF, W prefers FAF to FFF and C prefers FFA to FFF. That is, given that the other members vote for, it's better for you to vote against. However, consider the case where 2 vote For and one votes Against, either AFF or FAF or FFA. Can anyone do better by switching their vote given that the others don't? To be specific consider the strategy combination AFF. IF Auckland, voting Against, switches her vote and others don't, the salary increase will pass but all her electorate will know which we she voted, so she prefers not to change her vote in AFF. Would either Wellington or Canterbury want to change? Think about it. Consider Wellington, switching her "For" vote in AFF to Against means that the salary increase fails, whereas if she votes for it will pass, but now the electorate will never know! IF the electorate found out, then as above, she'd prefer to have the whole thing fail rather than have it pass and be seen to have voted for it. But under the new rules the electorate won't know!! Similarly for the strategy combination FFA. All three strategy combinations where only 1 member votes against and the others vote for, AFF,FAF, and FFA are all Nash equilibria. The problem is, how can the politicians coordinate on one of these? In a repeated voting situation they could perhaps take turns, keeping their own records of whose turn it is to vote against - otherwise there doesn't seem to be much else they can use for a focal point.