My apartment complex decided to show a movie via projector and sent out a poll with 6 options. We were asked to rank the options and told that the movie that won would be shown at the movie night.
If management only got one response, it would be easy to decide how to vote and which movie won. However, assuming there was more than one response, how should the winner be determined? And given that there will be other voters, what should your vote be? In this game, management has decided on the rules for voting (rank the 6 options) as well as the rule for which movie is selected based on those votes (which they did not tell us). The voters are then left to decide how to vote, given their guess of the rules.
The most likely way to score such a set of votes is that management could give a set number of points for each possible rank-position (i.e., 5 points for being ranked 1st, 4 points for 2nd, etc) and then take the movie with the highest sum. However, there's no reason they couldn't pick the movie with the largest number of times being ranked 1st, and then use the later places as tie breakers. And in general, there can be truly crazy sets of rules. For example, if one movie is far-and-away the favorite in general, we could handicap that movie by saying any vote for it is actually only worth half a vote (this seems ludicrous... but in auction theory counting a "high-valuation" bidder's bid as a fraction of their actual bid is a standard tool to design an "optimal" auction).
Depending on the rules, an individual voter has different incentives. Further, depending on their guess as to how everyone else will vote, they will have additional incentives. An important notion in game theory is a "Nash Equilibrium." A NE is a set of votes for everyone so no individual person will choose to switch their vote. So if you knew that everyone else was following the NE, there is no benefit to you from not following the NE. But there are a lot of assumptions that go into the NE including that it is unique, that there will be no collusion (lets both list our shared second-favorite movie as 1st), and that somehow there being a NE actually leads people to vote accordingly (here is the Wikipedia link on when that will happen).
Given all this complexity, you might wonder how anyone ever decides anything. In a world where so much is uncertain though, a lot of decisions could be the best. If I vote my genuine ranking, I'm at least giving my preferred movie it's best shot to be selected. I could vote strategically and rank my 3rd favorite as top because I think my least favorite movies are the most popular. I could also not vote at all because I think the effort involved in voting is more than the difference between the outcome when I vote or not. Which of these guesses of uncertainty is right is impossible to say until after the votes are in, which are then influenced by what everyone else is guessing to be the case.
Hopefully, this gives a taste of some of the difficulty both in designing the rules of a game, and the subsequent decisions by the voters. Early on in learning about game theory I was told "the devil is in the details," which I have found to be absolutely true. First-past-the-post voting seems sensible, until you realize the incentive issues when there are more than two choices.
Feel free to send me any follow-up game theory questions you have and I will do my best to get them answered!
If management only got one response, it would be easy to decide how to vote and which movie won. However, assuming there was more than one response, how should the winner be determined? And given that there will be other voters, what should your vote be? In this game, management has decided on the rules for voting (rank the 6 options) as well as the rule for which movie is selected based on those votes (which they did not tell us). The voters are then left to decide how to vote, given their guess of the rules.
The most likely way to score such a set of votes is that management could give a set number of points for each possible rank-position (i.e., 5 points for being ranked 1st, 4 points for 2nd, etc) and then take the movie with the highest sum. However, there's no reason they couldn't pick the movie with the largest number of times being ranked 1st, and then use the later places as tie breakers. And in general, there can be truly crazy sets of rules. For example, if one movie is far-and-away the favorite in general, we could handicap that movie by saying any vote for it is actually only worth half a vote (this seems ludicrous... but in auction theory counting a "high-valuation" bidder's bid as a fraction of their actual bid is a standard tool to design an "optimal" auction).
Depending on the rules, an individual voter has different incentives. Further, depending on their guess as to how everyone else will vote, they will have additional incentives. An important notion in game theory is a "Nash Equilibrium." A NE is a set of votes for everyone so no individual person will choose to switch their vote. So if you knew that everyone else was following the NE, there is no benefit to you from not following the NE. But there are a lot of assumptions that go into the NE including that it is unique, that there will be no collusion (lets both list our shared second-favorite movie as 1st), and that somehow there being a NE actually leads people to vote accordingly (here is the Wikipedia link on when that will happen).
Given all this complexity, you might wonder how anyone ever decides anything. In a world where so much is uncertain though, a lot of decisions could be the best. If I vote my genuine ranking, I'm at least giving my preferred movie it's best shot to be selected. I could vote strategically and rank my 3rd favorite as top because I think my least favorite movies are the most popular. I could also not vote at all because I think the effort involved in voting is more than the difference between the outcome when I vote or not. Which of these guesses of uncertainty is right is impossible to say until after the votes are in, which are then influenced by what everyone else is guessing to be the case.
Hopefully, this gives a taste of some of the difficulty both in designing the rules of a game, and the subsequent decisions by the voters. Early on in learning about game theory I was told "the devil is in the details," which I have found to be absolutely true. First-past-the-post voting seems sensible, until you realize the incentive issues when there are more than two choices.
Feel free to send me any follow-up game theory questions you have and I will do my best to get them answered!
Why is it optimal to discount a "high-valuation" bidder's bid, and what does it mean for a bidder to be "high-valuation"?
ReplyDeleteSo, here is the way I like to think about it:
DeleteImagine you're auctioning off a piece of artwork. There are two people who are bidding on it, someone who is very wealthy (W), and someone who is not (N). You would guess that W is willing to pay more than N. BUT, when you run a standard second-price auction, that means W will get the piece for the amount N is willing to pay.
So, because you believe W has a higher value for the artwork (meaning W is willing to pay more to get it), you discount W's bid. Now the auction should be more competitive because W will have to bid more aggressively to win the auction.
In summary, in this context "optimal" means you as the auctioneer choosing the rules make as much money as you can (on average). Any particular bidders value for the object is unknown to the auctioneer, but in game theory we usually assume some distribution for that value. If one bidder then has a valuation distribution which is bigger than the other, we handicap that bidder to level the playing field which gets us more money for the artwork.