Thursday, December 13, 2012

bank runs and time consistency

Diamond and Dybvig, in their famous model of bank runs, note that a bank regulator who commits to withdrawal freezes can forestall purely self-fulfilling runs; I no longer have to worry that my fellow depositors are going to withdraw first.  A few years ago, then Fed economists Ennis and Keister noted that a regulator may not be able to commit to such a policy, desiring in the midst of a bank run to allow some agents with highest demonstrated need to have privileged access to withdrawals, and that if agents anticipate this behavior the purely self-fulfilling run can still take place.  A somewhat fanciful solution is to note that, if somehow there were a liquid market for claims on bank deposits, people with high liquidity needs could sell their deposits to people with low liquidity needs; in fact, as long as the situation was expected to sort itself out within a week or two, the claims would probably trade very close to par, and, if this sort of solution seemed in the offing, there again would be very little incentive to trigger a self-fulfilling bank run.

How might one hope for such a liquid market to come into being?  The best idea I can think of is to make the bank into a market maker.  The bank could first, perhaps, be allowed to pay out to depositors at a rate conservatively estimated to be feasible if all depositors were paid; thus if depositors as a class are believed to be, with high probability, able to expect 25% of their deposits back in the event of failure, you allow depositors to claim deposits at 25 cents on the dollar.  That puts a floor on the market.  I'm hoping, though, that that would be irrelevant if the bank is also allowed to accept new deposits at a premium, and pay out on old deposits at a similar ratio.  For example, suppose a 10% rule: if you come to the bank right now and give it $10, your balance at the bank will go up by $11.  If the bank collects $180 this way, it is allowed to use that $180 to pay back old depositors at a 10% discount; i.e. if 30 people each line up to withdraw $9, reducing their account balance by $10, you pay out $9 each to the first 20 people in line.  Once the segregated, new funds are gone, the other 10 people can choose to get $2.50 or to wait until someone comes by to deposit more money.

On as rapid a basis as is practical, you could in fact change the 10%.  Again, if this is purely self-fulfilling, a 10% offer should bring in a lot of new money (which, among other things, would be somewhat costly to the bank).  On the other hand, perhaps in some situations in which there is real concern for the bank's solvency, you would have people standing around waiting to pounce on any new money that came in.  If the bank has a lot of new money, it could start cutting the 10% to 9%, and then 8%; on the other hand, if there are a lot of people waiting to redeem, you might start upping it to 11%, then 12%.

Again, if the bank is well-known to be solvent, this is all "off the equilibrium path", which is, of course, not to say that it's unimportant; the fact that it is credible is what keeps it from being needed.  I wonder whether it or something like it would be credible under some set of institutions that exists in the real world.

Wednesday, December 12, 2012

neural networks and prediction markets

I should start by noting that this idea is insane highly speculative, even by the standards of this blog.

I've been a bit obsessed with neural networks in the past two months. "Neural networks" constitute a class of models in which one seeks to generate an output from inputs by way of several intermediate stages (which might be loosely thought of as analogous to "neurons"); each internal node takes some of the inputs and/or outputs of other internal nodes and produces an output between 0 and 1, which then may be fed on as input to other nodes or may become the output of the network. With an appropriately designed network, one can model very complicated functions, and if one has a lot of data that are believed to have some complex relationship, trying to tune a neural network to the data is a reasonable approach to modeling the data.

This "tuning" takes place, typically, by a learning process in which the data in the sample are taken, often one at a time (in which case one would typically iterate over the data set multiple times), with some candidate network weights (typically initially random) are used to calculate the modeled output from the given input; the given output (associated with the data point) is then compared to the modeled output. The network is designed such that it is relatively efficient to figure out how the modeled output would have differed for a slightly different value for each weight — one effectively has partial derivatives of the output with respect to each weight in the network — and a certain amount of crude nudging is done to the entire network, more gently if the modeled output was already close to the actual target output, more forcefully if it was farther away, but generically in the direction of making the network predict that point a bit better if it sees it again.

One of the problems with neural networks — and one I'm going to exacerbate rather than ameliorate or exploit — is that the resulting internal nodes typically have no intuitive meaning. You design your network with some flexibility, you go through the data set tuning the weights to the data, you come back with perhaps a decent ability to predict out-of-sample, but if someone asks "why is it predicting 0.9 this time?" the answer is "well, that's what came out of the network", and it's typically hard to say much else. They may still be useful in somewhat indirect manners as a tool for actually understanding your data generating process, but even if they can perfectly model the sample data, they at best provide descriptive dimensional reduction from which direct inference is essentially impossible.

Now, I've been interested in prediction markets for longer than I have neural networks, though perhaps less intensively, especially in the recent past. Prediction markets typically are intended to aggregate different kinds of information into a sort of best consensus prediction of, typically, the probability of an event. If different people have different kinds of information, then combinatorial markets are valuable; if I have very good reason to believe that the event in question will occur provided some second event occurs, and someone else has very good reason to believe that the second event will occur, then it may be that neither of us is willing to bet in the primary market, but if a second market is set up on the probability of the secondary event, the other guy bids up the probability in that market and I can bid up the primary market, hedging in the secondary market. (A true combinatorial market would work better than this, but this is the basic idea.) In principle, a large, perfectly functioning combinatorial market should be able to make pretty good predictions by aggregating all kinds of different information in the appropriate ways, such that the market itself makes predictions that elude any single participant. (cf. a test project for this sort of thing and some general background on that market in particular.) The more relevant "partial calculation" markets are available, the better the ability to aggregate to a particular event prediction is likely to be.

There would be some details to work out, but it seems to me one might be able to create a neural network in which the node outputs are not specific functions per se of the inputs, but are simply the result of betting markets. Participants would be paid a function of their bet, the output, and the inputs, as well as the final (aggregate network) realized ("target") result. If you make a bet on a particular node, the incentives should be similar to the "tuning" step in the neural network problem: you should be induced to push the output of that node in such a direction that the markets downstream from you are likely to lead to a better overall prediction.

It's quite possible there's actually no way to do this; I haven't worked it out in even approximate detail. It's also possible that it is possible in principle, but there would be no way to get intelligent participation because of the conceptual problem with neural network nodes that I mentioned before. If it did work, in fact, I suspect it would do so by means of participants gradually attributing meanings to different nodes. If this could be made to work in something like the spirit in which neural networks are usually done, this would improve slightly on the combinatorial markets in that
  1. The intermediate markets would be created endogenously by the market; that is, something the market designer didn't think of as a relevant partial calculation might end up being ascribed to a node because market participants had reason to believe it should be; and
  2. These intermediate markets may not need a clear (external) "fix"; that is, some events are hard to define, but as long as participants have a sense of what an event means, it doesn't have to be made precise.
Let me clarify that second point with an example: perhaps the main event is the Presidential election, and one partial result is whether Romney wins the second debate. One might in principle be able to make this precise — "... according to a poll by CNN immediately following the debate," etc. — but if the payout isn't officially determined by that, and is only officially a function of observable market prices and the final result (who wins the election?), it can take on a subjective meaning that never officially has to be turned into something concrete. Indeed, it may be that different agents would believe the node to mean different (probably highly correlated) things; perhaps some people think it means Romney wins the second debate, and others think it means he wins "the debates". Only the result of the election needs to be formalized.

Tuesday, December 11, 2012

football playoffs

Major college football has been very gradually moving toward a playoff system, with many fans clamoring for a quicker move to a larger playoff. Proposed playoffs are almost always single-elimination; insofar as one is more likely to accurately determine which team is the best on the basis of more information than less, allowing a single loss in an expansive post-season playoff to eliminate a team from contention, with the playoff including a number of teams that lost two or even three games during the regular season, amounts to throwing away information, and makes it less likely, not more, that the winner of the playoff will actually have been the "best" team on a season-wide basis. A compromise idea I've played with is privileging the teams that seem, on the basis of the regular season, most likely to be the best, but allowing lower-ranked teams into the playoff under less advantageous terms; a top team would be permitted to remain in the playoffs after a loss, while a lower-ranked team would not. A couple years ago, I suggested a playoff system to my brother and he informed me that the playoff system for the Australian Football League is essentially what I had proposed. A couple weeks ago, CNNSI assembled a mock committee of actual people who might be on a real committee selecting a college football playoff as will happen in a couple of years. If I use this to seed an Australian Football League style playoff, I get a bracket like
Notre Dame
GeorgiaTexas A&MTexas A&M
Texas A&MOregon
FloridaNotre DameAlabama
LSUStanfordNotre Dame

with game results projected in the first three rounds in part to clarify the structure of the bracket; two games in the first round pit top 4 teams, and Florida and Notre Dame, by virtue of losing those, are sent across the bracket to their second-chance games against lower-ranked teams that entered on single-elimination terms.  The semifinals here feature Oregon against an SEC team and Notre Dame against an SEC team.
One feature of playoffs in all four "major professional sports" in the US(/Canada) is that the leagues consist of two "conferences" and that the playoffs keep each "conference" separate; the playoff systems thereby create a championship match (or series) that has one team from each half of the league, rather than seeking straightforwardly to pair up the top two teams. In baseball the two halves of the league play with slightly different rules, and in basketball and hockey they have a certain geographical logic, but in the NFL in particular the division is entirely historical; having "AFC champions" and "NFC champions" I suppose gives the team that loses a little bit more euphemistic title than "Super Bowl loser" — and perhaps even a team that "has been to n of the last m AFC championships" even feels it's accomplished more than a team that "has been to n of the last m quarterfinals". It feels a bit hollow to me, and I'd just as soon see a single tournament. In the regular season, at the moment, each team plays 12 games within its conference and only 4 against the other conference; this could be modified, but as a first step, perhaps we should take six teams from each conference into the playoffs, seeded separately, and have them play
The top two teams from each conference don't get byes — at least not at first.  They are placed in a four-team single elimination tournament of which the winner gets a double bye, both the third and the fourth round.  The other eight teams play their own games, of which the winners "catch" the three teams that lose from the top-four tournament; the first-round losers play 4/5 winners in the second round, and the second-round loser plays the "champion" of a four-team 3/6 tournament in the third round, as the winners of the 4/5-1/2 games play each other.  After round 3 there are 3 teams left: the undefeated 1/2 team, and two teams that are each either an undefeated 3–6 seed or a one-loss 1/2 seed. If we let the latter teams play in round 4, the winner of that gets a(nother) shot at the undefeated 1/2 team. A 3–6 seed can win the championship, but needs five wins in a row, with a few of them probably top seeds; a 1/2 can win with a loss, but it requires that the team go 4–1. Perhaps a different visualization would be useful; one spot in the Super Bowl is filled by a four-team single-elimination tournament, and the other is filled by
AFC 1w(loser)w
NFC 1w
AFC 3ww
NFC 3w
AFC 1(loser)ww
AFC 4w
NFC 1(loser)w
NFC 4w
Note that the 1/2 games, listed twice in the table, aren't played twice; a w denotes that the winner of the previous round advances to that spot, while (loser) denotes that the loser from the previous round advances to that spot.

As a couple final remarks,
  • The NFL playoffs as currently constituted are, as far as I know, unique in that there is not a fixed "bracket"; a team that gets a bye into the second round doesn't have a particular game of which it plays the winner. What I have produced here is a more traditional "bracket" in that sense. I'm not necessarily opposed to the NFL's system in that regard; this is just what I did.
  • Major college football does have a number of "conference championship" games, all of which take one team from each of two "divisions" of a conference, rather than taking the top two regardless of division. This year Ohio State, because of previous misdeeds, was ineligible to play in the championship game of its conference, but was declared the champion of its division; the team in its division that finished highest in the standings while also not being under instutitional sanctions went to the championship game instead. There was some lack of clarity, midway through the season, as to whether Ohio State would be allowed to be the "division champion"; it seems to me that the decision that was made vitiates much of the reason for the structure of the championship game. If the point isn't to match the two "division champions", it should be to match the top two teams. The asymmetric schedule makes a case for some preference toward having teams from different divisions, but in this case the team that went in Ohio State's place was a full two games behind a team from the other division that didn't make the championship game and that would have seemed to have a rather better case for being invited.
Update: Let's do a mock bracket.  (Updated Dec 31.)  I'm adopting a proposal by SI writer Petere King that division champions be guaranteed a playoff spot, but not a top 4 seed. I'm also, naturally, making some guesses about how the rest of the season will play out; that said, my teams are
New EnglandSan Francisco
HoustonGreen Bay
which might lead to something like
DenverSan Francisco(New England)New England
San Francisco
New England
New England
HoustonHoustonGreen Bay
Green BayGreen Bay
San Francisco
New England
and the winner of the New England vs. Baltimore match gets to play San Francisco in the Super Bowl.  To clarify, a team in parentheses lost its previous game in order to land in that spot.

I will note some features of the bracket that should perhaps have been mentioned before (they aren't specific to this simulation):
  • As long as no lower seed beats an upper seed, teams from the same conference won't play each other until at least the third round; any such matchup must follow a team beating a seed at least as high as itself.
  • Two teams will not play each other a second time — there will be no "rematches" — until at least the fourth round (as there is in the mock bracket), at which point there are only three teams left and you're running out of ways to avoid them.
Both these points are to say that I've laid it out to create a lot of "mixing", so that teams that are a bit hard to compare before the game — they're in different conferences, so they have few common opponents, and they haven't played before in the tournament at least — are more likely to be paired than teams that are more readily ranked relative to each other.

Wednesday, November 7, 2012

general equilibrium: campaigns and voting systems

Previously I extolled the virtues of approval voting, in which voters vote for as many candidates as they wish, with the highest vote total determining the winner; as I indicated, it is in many ways a substantial improvement on mandatory bullet voting, in particular in situations with large amounts of information (e.g. pre-election polls) and with rational agents, in which case it is in some sense optimal.

I've been thinking in the past day or two about the idea of allowing up to two votes (for a single winner), but not more than that; just as the single-vote system results in two major candidates, allowing two would by and large result in three or four major candidates, which might be less informationally demanding on voters than if there are a potentially unlimited number.  (An alternative is to have a primary of some sort to fix the number of general election candidates; in one of the environments I've mentioned, where one supposes that it is straightforward for voters to change their votes over a period of time with running totals publicly available, one way of doing this would be similar to a number of TV contests ("Survivor", "American Idol"), in which perhaps only the top 20 vote recipients on January 1 are allowed into the race, with the lowest vote getter eliminated every two weeks until you're down to, say, four.)

What I want to highlight this morning, in the general situation in which there are more than two serious (or reasonably potentially serious) candidates, is how this would affect campaigning; in particular, I would expect less "negative" campaigning.  If I'm candidate A running against candidate B with no other real opposition, it can make sense to devote a lot of resources to searching out the dumbest thing candidate B ever said, or even the thing candidate B ever said that is most amenable to being misconstrued; similarly, any unforced errors by candidate B — generically, anything that makes candidate B look worse — benefits me. To some extent this will still be true when candidates C and D are also reasonably in the mix, but it's not the case to nearly the same extent; even if I get a picture of candidate B paying a prostitute, if I don't have much to say for myself, there's a decent chance candidate C or D picks up his support.

I put "negative" in quotes because some of what is sometimes given that label is quite relevant to voters' decisions; some commentators seem inclined to label and ad that says "candidate B voted for bill X" as negative if run by candidate A and as positive if run by candidate B.  Voters are being asked to choose between options, differences between those options are relevant, whether they guide voters with different preferences in different directions, or whether they guide almost all voters in the same direction.  That said, some of the worst of campaigns — the seemingly deliberate misconstruals, or most brazen attempts to make the other guy seem alien, like "not one of us" — would become less relatively profitable if there were more candidates; indeed, on at least a rough basis, I imagine that having three opponents would make investing resources extolling your own virtues about three times as relatively valuable, compared to making the other candidate look bad, as only having one.

Tuesday, November 6, 2012

graphical representation of the presidential election

I hope that there is displayed below a graphic displaying states that would be anywhere near in play that have been called in tonight's presidential election. It's possible that permissions are wrong and that I'm seeing it but nobody else is. Let me know in the comments if this is even less intelligible than I expect. Note that the spirit is the same as in the earlier post; a simple count of electoral votes at any point in the middle of the evening is indicative more of which states' polls closed early than of an actual clear lead by one candidate or the other. In this display, if you start seeing red incursions in the blue territory or vice versa, that's your signal that one candidate or the other has done something clearly demonstrative of a likely electoral college win.

Monday, November 5, 2012

keeping score with the electoral college

Here's how I would watch the results come in if I were to actually do so:
  • If Obama wins Florida, I don't see him losing; similarly, if Romney wins Pennsylvania, you can go to bed early. (This isn't to say that it's not possible that Romney would win Minnesota or Obama would win Missouri, just that a lot of what we don't know about one state is correlated with what we don't know about others. If Obama wins Florida, he'll win Minnesota.) Both states close their polls at 8:00 Eastern Time, and in the case of a modest landslide (of the order Obama won 4 years ago) they would likely be the first clear indications of such.
  • These are the swing states to watch, with the number of electoral votes for each and the poll closing time, converted to Eastern Time:
    Virginia137:00If it is going to remain in any suspense well into the night, Obama has 236 electoral votes not listed at the left, and Romney has 235. The fact that Romney will be up 64-3 at 7:45 or so will be meaningless; I'd start from 236-235 rather than 0-0, adding points for swing states, and shifting points for upsets. If Romney does win Pennsylvania, add 20 to his total and deduct 20 from Obama's. If the overall race is close, I don't expect any deducting; you'll just, for example, add 13 to whomever wins Virginia. If there's no deducting, then whomever hits 270 wins. If Romney gets 35 from these states, or Obama gets 34, before any upsets are announced, I think upsets are pretty unlikely.
    Maine 2nd district18:00
    New Hampshire48:00
  • For example, if it's getting close to 8:00 and Virginia is called for Romney, your tally is now 248-236 in his favor; if Ohio is then announced for Obama, the total is 254-248 in his favor; if Florida is then called for Obama, adjust it to 283-219, and expect that the binary result is in little remaining doubt.

Thursday, October 25, 2012

Congressional apportionment

The number of representatives in the House to which each state is entitled is supposed to be proportional to the population of each state; however, it is required to be an integer for each state as well.  For the last 8 House reapportionment cycles, the mechanism used has been one that gives an apportionment that has the smallest possible standard deviation of the logarithm of the population per district; for each state, figure out how many people there are per congressional district and find the logarithm, then look at the standard deviation over the 50 states.  If you weren't forced to use integers, you'd like to make it 0; instead, you make it as small as possible, conditional on there being 435 districts allocated as integers.

The 2000 apportionment created perhaps even more fuss than usual because Utah and North Carolina were so close to each other in terms of which was in line for the 435th representative and which the 436th.  This is related to the fact that the standard deviation of the log district size was quite high relative to what it would have been if there had only been 434; if North Carolina hadn't received the additional district, it would have been more in line with the other states, and adding one to Utah after that would have improved things slightly.  It might have made sense, on these grounds, to shrink the House a little bit to make district sizes a bit more uniform.

Now, if there were one congressman per person, the restriction to integers wouldn't be an issue; all districts would be the same size, and the standard deviation would be 0.  Even if there were say 3000 congressmen, it should be intuitively clear that the stage in which one approximates a perfectly proportionate definition with integers is going to involve much smaller (proportional) changes than for 435, which is to say that there is, on a large scale, going to be a general downward drift, with the standard deviation going approximately as the inverse of the number of congressional districts.  Adding representatives is obviously costly at some point, primarily in the functioning of the legislative body (pecuniary staff costs and so on are presumably dwarfed by this); if you're simply going for the minimum global value, but you want a sane minimum, you have to add a term that grows in the number of representatives.  There's a reason to make this inversely proportional to the number of states, i.e. to add a cost per (congressmen per state) rather than per congressman per se; the plot below shows the standard deviation of log district size plus 1% times the number of congressmen per state, viz. the standard deviation plus House size divided by 5,000.  Aside from seeming like nice numbers to a species with ten fingers, it happens to suggest a House size near the size of the actual House of Representatives.  The exact minimum is 413, but any number between 410 and 430 is fairly close.

As it happens, the optimum for the 2010 allocation is a bit bigger, but similar; the minimum is at 424, but anything from 421 to 429 is close.  If the size of the House were reduced to 424, California and Texas would each lose 2 of the 11 representatives; Florida, Georgia, Minnesota, New York, Pennsylvania, South Carolina, and Washington would lose the other 7.

Wednesday, September 26, 2012

income shares and productivity

Even more than usually for this blog, this post is my way of filing away the merest germ of an idea where I might be able to find it again and perhaps turn it into something some day.

The Cleveland Fed notes that one of the most famously stable macroeconomic ratios has broken out of its long-term range; the share of income going to labor has been dropping, particularly over the last couple of years. Part of this may be accounting — I'm going to want to think first about what is actually being measured and how it relates to theoretical constructs — but it's also interesting to note that when labor share broke above its long-term channel, in the very late sixties and early seventies, that it coincided with a secular drop in productivity growth. Combine this with the stylized fact that the benefit from innovations tends first to go to the developers (in a broad sense) of the innovation, then to investors who use the innovation, and, after a decade or two, ultimately to consumers, as the rents previously enjoyed by the investors are competed away, and it seems like a causal link might be plausible, in which case perhaps the fact that income to capital (in a time of 0 interest rates! Again, I need to be sure what we're measuring) is running in front of income to labor is an early indicator of innovations that will drive economic growth in the next two decades.

No doubt this seems like an optimistic spin, and is probably more speculative even than most things I post here.

Sunday, September 16, 2012

extending the model: salutary log-rolling

A few weeks ago I wrote about voting for a single winner; I'd like to extend this to more general situations, in which we can amplify the "intensity" property that was obtained in the special case in which there was no Condorcet winner. This is related somewhat to the idea of storable votes, which has worked well in a lab with human subject experimentation.

Let's start with an example, in which we have three misanthropes.  The three misanthropes are going to vote on three binary issues: do we kill misanthrope 1, do we kill misanthrope 2, and do we kill misanthrope 3?  (The killing is not done until all three decisions have been made; misanthrope 1 gets to vote on all three questions, regardless of the answer to the first question.)  Each would prefer to kill each of the others, but would much more strongly prefer not to be killed himself.  By way of usual, issue-by-issue voting, each would be killed; intensity of preference can't be accounted for.  However, by considering this a choice of 8 possible outcomes, we can make it a (no longer binary) single-issue vote; it has no Condorcet winner, as the result "don't kill anyone" beats any result that kills at least two people, each of which beat two of the results in which one person is killed, each of which beats the "don't kill anyone" optimum.

Now, this is actually a bit simpler on some level to analyze in the many-voter limit; let's suppose there are three factions of approximately equal sizes, each voting whether or not to annihilate each faction.  Now, with approval voting, in equilibrium everyone votes against every outcome in which they would die and votes for every outcome in which they survive and at least one of the other factions is annihilated.  If it looks like there is a good chance that they will be annihilated, they will vote for the outcome in which nobody is killed as well; if they believe themselves to be practically safe, they might vote against this outcome.  In equilibrium, the outcome in which nobody is killed will usually win, with "usually" getting more and more probable as the importance to each person of surviving becomes more important to them relative to their own bloodlust.

In situations with many votes, some of which may not be binary, this can get much more complicated.  The ability of the system to take account of intensity improves as the number of votes aggregated increases; we might, for example, want Congress to vote at the end of the session on every bill brought to the floor in the previous two years.  This, however, would be entirely impractical, even supposing none of the votes was on something of particular urgency.  (Say, a declaration of war.)

What might make it less impractical is the fact that the equilibria will be unaffected by the addition of irrelevant alternatives.  In the case of the misanthropes, the equilibrium consists of a large probability of nobody being killed and small, equal probabilities of one person being killed; if we remove some of the other four options from the set of allowed possibilities, the equilibrium stays the same.  In the case of millions or billions of combinations of possible vote outcomes, if there is a Condorcet winner, then if that Condorcet winner could be found, then if that were offered along with any smallish set of other combinations of outcomes to voters to then vote, based on approval voting, for the whole combination, that result would win the vote, regardless of what the alternatives were.  On the other hand, if the Condorcet winner were not included in the set of options, adding it would upset the equilibrium that was obtained from the set that excluded it.  Thus we might hope that allowing some means of offering competing slates of results to "challenge" the current winner of the vote would allow the Condorcet winner to be found; as long as someone is clever enough, at some point along the way, to get that option nominated, it will win, regardless of what other nominees are in place.

It seems then that a good first guess would be to do the vote in the traditional way — have each voter vote issue-by-issue, but use approval voting for natural ternary options. (If there are two competing jobs bills, and nobody thinks they should both happen, combine into a single vote "do jobs bill A, do jobs bill B, or don't do either".) Take the resulting combination of results, and allow the nomination of challenges to it.  If the bill on which I feel the most strongly went against me, and I think that a lot of people would be willing to change that result if we also changed some other set of results on which my opponents feel more strongly and I feel less strongly, it might make sense to nominate that.  (For example, after we all vote to kill everyone, I can propose letting everyone live, which would get everyone's vote against the first equilibrium; I prefer letting myself live very strongly, and others oppose it weakly, so I propose that we change that in the direction of my preference while changing the other results against my preference.)  If more than about 10 or 15 combinations have been proposed, you can drop those that are getting the fewest votes, allowing for new nominations of new potential combinations while keeping the number of combinations under consideration manageable.

In the spirit of this blog, there is probably a lot that I haven't worked out; in particular, for certain common classes or distributions of preferences, it seems likely that there is some way of further tamping down on the informational complexity I'm asking voters to manage.  In particular, perhaps there's a way to solicit from voters their preference intensities on different issues so as to compute likely alternative combinations of results that can be placed into nomination for that revision phase.

Thursday, August 23, 2012

voting for a single winner

In a situation in which people are voting for one outcome out of several choices, there is some question as to how we should go about aggregating people's preferences.  Elections in the United States typically ask each person to vote for one candidate, and the candidate with the largest number of votes wins.  This leads to phenomena like "vote-splitting", where two similar candidates may each get 30% of the vote while a third candidate gets 40%; if either of the similar candidates had dropped out, the other would have won.  This feels wrong, but it turns out that it is generically hard to avoid this problem; in particular, as wikipedia explains it,
In short, the theorem states that no rank-order voting system can be designed that satisfies these three "fairness" criteria:
  • If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
  • If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
  • There is no "dictator": no single voter possesses the power to always determine the group's preference.
The second condition is essentially this vote-splitting criterion.  Note that the theorem worked with more flexible ballots than we usually use — Arrow would ask voters to rank all of their alternatives, rather than only state a first choice — and supposes that the voter reports preferences honestly instead of strategizing.*

In somewhat recent work, it has been noted that in practice preferences might be likely to be constrained somewhat; if there are five candidates, for example, it might be odd to see a voter whose first choice is the most conservative candidate and whose second is the most liberal.  If you rule out preferences of this sort, then you can do much better; in particular, taking the ordered ballots from each voter, you can infer, for each pair of candidates, which candidate would win in a two-candidate "run-off", and if one candidate beats all of the other candidates in such pair-wise elections, it makes sense to call that candidate the winner.  This system satisfies all of Arrow's criteria, except that it can sometimes fail to choose a winner (if candidate A beats B, B beats C, but C beats A in pairwise elections).  When it does choose a winner, we call that winner the "Condorcet winner", and in such situations it tends to be fairly attractive — for example, in a lot of cases it helps voters find compromise candidates who might be ignored in a single-preference voting system.  It turns out that it is necessary for some voters to have "odd preferences" in order for there not to be a Condorcet winner; in fact, among voting systems that satisfy Arrow's criteria for some set of allowable preferences, this system is the most permissive.

As an example, suppose about 40% of the voters prefer the liberal candidate, about 40% prefer the conservative candidate, and 20% prefer a "moderate" candidate.  Presumably, in a two-candidate race, the moderate candidate would beat either of the others.  In a three-candidate race, if it becomes conventional wisdom that the moderate candidate can't win, voters with a single-preference vote strategically shift toward whichever of the "major" candidates they prefer.  Note that, if all voters believe that the conservative candidate can't win, then the moderate candidate will pick up the 60% vote and win.  The main problem with the single-preference vote is that there are self-fulfilling beliefs that lead to other outcomes.  Asking each voter for a ranking of candidates allows the voting system to work out the better outcome itself.

It seems attractive, then, to get people away from single-preference voting, and the real enthusiasts of voting reform tend to be big proponents of ordered ballots.  They are used successfully by Cambridge, MA, for its elections for city council and school board, but there are much less radical changes that can yield a lot of benefits relative to our current single-preference system; in particular, instead of saying "vote for one candidate", we can say "vote for as many candidates as you like".  If there are five candidates and you really dislike one of them, and don't have strong preferences among the other four, you can effectively vote "against" that one candidate (by voting for all of the others).  The system is, however, typically called "approval voting", with the idea that you indicate which candidates receive your approval.  In our above example, if it is believed that the moderate candidate has no chance, then the moderate voters can continue to vote for their preferred candidate, even as they also vote strategically for one of the major candidates; if it looks like the liberal candidate has a good chance of winning, then the conservative voters, too, will vote for the moderate candidate in addition to their own, and, conversely, if it looks like the conservative candidate has a good chance of winning, the liberals will vote for the moderate as well as the liberal.  There is no set of common, self-fulfilling beliefs in which the moderate fails to win.

A lot of game theory generally and voting theory in particular works in these "full-information" (or "almost full-information") environments, in which all voters have the same beliefs about what the outcome of an election will be, with these beliefs clustered very tightly around the actual outcome; in an election with 1,000,000 votes cast, people might not be quite sure exactly how many votes each candidate will get, but they would be able to guess to within about 1,000 votes.  This is clearly unrealistic; indeed, another problem with the single-preference ballot is that much of the primary system is spent trying to influence beliefs about which candidates are likely to win, and voters spend a lot of their energy trying to form these beliefs and strategizing around them.  Approval voting reduces this problem, but doesn't eliminate it; it is still not that unlikely that, even if there is only one "equilibrium" (i.e. set of self-fulfilling common beliefs) and it has a good outcome, voters with different expectations would find themselves surprised by the outcome and, after improperly strategizing, would effect the wrong result.

Consider, then, an environment in which it's practical to let people change their vote, and in which releasing running vote totals doesn't compromise privacy that we want to protect.[2] I believe in this context -- essentially a full information context -- that approval voting has a unique equilibrium selecting the Condorcet winner if there is one,Update:this turns out to be false. and that it does not have a deterministic equilibrium (i.e. one in which people are practically certain who's going to win) if there is no Condorcet winner.  In this extended-vote format, I would expect the result to stabilize when there is a Condorcet winner, and, when there isn't one, for the "leading candidate" to switch from one candidate to another among a set of candidates with fairly broad support and good compromise qualities, similar to Condorcet winners.  I therefore propose:
  1. Allow voting for a fixed period of time, in which voters are free to vote for as many candidates as they want, and may vote or rescind their vote for any candidate at any time in this period.
  2. Maintain a public tally of the number of votes for each candidate for the bulk of this period.
  3. Stop releasing the public tallies shortly before the close of the voting period.  (I'm imagining something like 15 or 30 seconds in the context of a vote for Speaker of the House, and certainly much longer in anything like a general election; long enough for voters to give a little bit of last-minute strategic thought and to enter their final votes.)
  4. In the event of a tie, the last votes to change lose priority; thus if two candidates tie, and they were tied before the last vote change, but the penultimate vote change was for or against one of them, that penultimate change is disallowed.
The purpose of 4 is partly to give a definite winner, but also to create a small incentive to vote early.  The purpose of 2 is to allow coordination, so that the result should end up in an equilibrium (if there is one) or a near-equilibrium; if the voters are about to settle on a bad outcome because they have mistaken beliefs about who is winning, they should see that in the early returns and be able to change their votes in light of the new information.  (1 also can be viewed as having this purpose, insofar as single-preference voting tends to have bad equilibria; single-preference voting would be enough to thwart the rest of the design, which would simply lead voters to lock in to some equilibrium, but quite possibly the wrong one.)  I would presume, then, that this system leads to a Condorcet winner where one exists; there remains the question, though, of choosing a winner when there is no such candidate.  The purpose of 3 is to allow the final vote to be taken under some kind of fairly shared and fairly accurate public belief about who the most popular candidates are; rather than having people race to change their votes at the last minute (again, also part of the reason for 4), I want the decision to be made on intensity of preference: e.g. which do you prefer, your second choice of the 3 viable candidates, or a 50% chance of your first choice and a 50% chance of your third choice?  On some level, I would like to take account of intensity of preferences more generally, but more generally attempts to do so tend to lead to strategizing that undermines the attempt; in the case in which there is no Condorcet winner, the proposed program is a way to do so that should be robust to attempts at strategy.

* It's worth noting that the three criteria are related to strategy-proofness in an important way. If everyone knows what votes everyone else is casting, any system designed such that the voter will report his true preferences regardless of what everyone else's vote is must obey the first two of Arrow's criteria. In more general settings it is convenient to suppose that the voting system itself does any necessary "strategizing" — if a voting system would reward a voter with one set of preferences for reporting a different set of preferences, the system should simply treat a report of the first sort as a report of the second sort, so that the voters can simply report honestly.

[2]The former is certainly more straightforward where there are no secret ballots; in the case of the latter, I'm imagining you watch someone walk into a voting booth and see a candidate's total go up as they vote and infer how they voted. In a large election with computerized voting, you could allow the computer to keep track of your current vote, allowing you to change it; then you just have to trust that this will stay anonymous. Also in a large election, many people will be voting at about the same time, so the privacy concern is reduced; in addition, instead of releasing vote totals continuously, you could release them every few minutes, for example. Finally, I would note that a system of daily tracking polls to some extent effects a system like this; indeed, if we could just get tracking pollsters to ask "Is your support strategic, and, if it is, what candidates do you prefer to the one you're voting for?", most of my idea would be implemented.

Thursday, June 28, 2012

college football

For something (almost) completely different, I'd like to make some proposals regarding playoffs for the top division in college football.

My objection to an expansive playoff is primarily that the best team doesn't win every game, even insofar as there is such a thing as a best team; if #16 beats #1 in the first elimination game, I'm going to be wondering whether the winner of the tournament really is better than the team that lost that first match.  Of course, polls and selection committees aren't perfect, either, which is at least part of why playoff systems are popular; if there's a legitimate case to be made that #3 is likely to be better than #1 or #2, let's let them play each other and find out.*

Part of the difficulty of comparing teams on the basis of their 12 games each is that there is so frequently little overlap among those teams' opponents.  Not only did Oklahoma State not play Alabama last year, but no team Oklahoma State played played Alabama last year.  There is an increasing tendency for teams to play almost all of their games in their own conference; this is sometimes hailed as a good thing, typically because the best teams tend not to play very good teams in their non-conference games — which makes it even harder to compare good teams in different conferences than if the occasional shared opponent were at least good. It would be better to have stronger schedules that weren't quite so incestuous.

On the other hand, in college basketball, some of the lower "mid-major" conference teams have complained that they have a hard time getting teams from the top conferences to play them precisely because strength-of-schedule is so emphasized; the top teams don't want to play a team that might hurt their strength-of-schedule metrics, so those teams never get a chance to develop the resumes that would make them look strong.  It would be nice if incentives induced teams to give the outsiders a chance to show what they have, but without creating too many rewards for beating up truly weak opponents.

Accordingly, here is my proposal:
  1. A six-team tournament, comprising
    • the four teams believed by some reasonable process (e.g. a committee) to have the best claim to being the best team in the country
    • The next two teams selected by a process that emphasizes schedule strength and diversity
  2. Teams 5 and 6 are selected according to the following ratings (probably modified for teams that play more than 12 games) that awards points to each team for each game between division I-A football teams:
    • For teams i and j, let nij be the smallest number of games required to link them; thus nii=0, nij=1 if i plays j, nij=2 if i did not play j (and i is not j) but they played a team in common, etc.
    • Team i gets 1/8nij points for any game that team j wins
    Thus you get 1 point for a game you win, 1/8 for a game an opponent of yours wins, etc. — but you can only get credit for a game once
The rating system gives too much credit for strength of schedule, but only the two "incentive spots" in the playoffs are awarded on that basis; they won't be bad teams, and they won't get to the semifinals without beating a top 4 team, so they won't screw up the integrity of the playoffs, they just provide an inducement to athletic directors not to try to schedule too many weak teams.  On the other hand, if a mediocre team is being ignored by all the good teams, and is going to end up 8-4 against a weak schedule, then adding that team to your schedule is going to do you more good than adding a 6-6 team with a tougher schedule against teams that some of your other opponents are also playing; you move those 8 wins to a lower nij, as well as the wins of their opponents.  The general equilibrium effect, then, is actually to improve the ability of the committee to select the top four teams; when scheduling has been done knowing that, whatever else happens, a team with a good rating can snag one of these two incentive spots as a backup, that scheduling will result in more Oklahoma States scheduling decent common opponents with more Alabamas, increasing the likelihood that the teams that make the semifinals include the best one or two teams in the country.

*The race isn't always to the swift nor the battle to the strong, but that's usually the way to bet.

Saturday, June 23, 2012

credit ratings and regulation

The ratings agencies were comically* inept at rating credit tranches of, well, everything five years ago, and are probably not much better now.  Even when rating boring old corporate and sovereign bonds, they've typically been at best a lagging indicator of events; wherever there's a quasi-liquid market indication of whatever it is the credit ratings are supposed to mean, the market has typically priced in an increase in credit risk long before the ratings get around to doing downgrades.  It may be that they provide some value in the case of new issues; their ratings of bonds from companies that haven't previously issued bonds before may impound some research that would be costly for a lot of bond investors to replicate.  It seems mostly, however, as though 1) the credit ratings are inadequate, 2) the market knows it, and 3) the only reason the credit ratings carry any importance at this point is because of capital regulations — various banks and insurance companies are required to hold a certain amount of their assets in securities with various ratings, or else are required to maintain leverage ratios that depend on the securities' credit ratings.

Many large investment banks spent much of the middle of the last decade specifically constructing securities that they knew would get high credit ratings, but with as high a yield as possible, so that regulated entities would want to buy them. The very fact that they had high yields should have been a tip-off to regulators; the market is not always efficient, but there are certain ways in which it tends not to be too inefficient, and if a liquid marketable security, in times of low credit spreads and rising stock markets, has a high yield, it might be worth the regulators' wondering whether the market collectively knows something that a simplistic model has missed.

If we really think a fund of some sort should only, by regulation, invest in securities with some maximal leverage, perhaps it would make sense to impose (say) a 75% tax on gains above some maximum rate of return; perhaps 2 percentage points above ten or thirty year treasuries would be "investment grade".  This idea isn't quite just motivated by a general belief in a risk/reward tradeoff, though it's perhaps more closely related than should be made distinct: the idea is to get these investors in the mindset that, conditional on such-and-such a return, we're optimizing risk, rather than vice-versa.  This shifts the competition for pension-fund dollars from price-competition to quality-competition, in essentially the classical way in which this is done; by making it impossible for a creator of securities to compete by offering higher yields, I'm hoping they will be compelled to compete by offering more safety — safety as determined by what the market knows now, and not just information that is old enough to have finally been acknowledged by the ratings agencies.

* I have a sick sense of humor.

Wednesday, June 13, 2012

ceteris paribus and causality

One of economists' favorite latin phrases is "ceteris paribus", meaning "the rest unchanged"; the idea is that we're looking at the result of changing one thing while leaving everything else the same to isolate just the effect this one change has.  The moderately perceptive student notes that at some level this is impossible, or in any case is at variance with the rest of what is being taught.  For example, Y=MV, where Y is (nominal) income, M is the money supply, and V is the velocity of money.  If I double the money supply, either income has to increase, or velocity has to decrease, or both; it is not possible to double the money supply but leave everything else the same.

"Everything else the same" is approximately the idea behind the partial derivative from multivariate calculus.  If I have a function f(x,y), and I ask for ∂f/∂x, I'm asking for the change in f in response to a small change in x with y held constant.  Thermodynamics, in physics, is the only context in which I've seen the ceteris paribus problem formally attended to; for an ideal gas, for example, PV=nRT, and entropy can be written as a function of any two of P, V, and T; specific heat at constant volume CV is T(∂S/∂T)V, that is to say with volume constant and P changing as T changes, while CP=T(∂S/∂T)P is the corresponding expression for constant pressure (with V changing linearly with T). The two can be related; CP-CV=VT(α2/κ) where α=(1/V)(∂V/∂T)P and κ=-(1/V)(∂V/∂ P)T are the volume coefficient of expansion and the isothermal compressibility of the gas, respectively. This is related to the Slutsky equation from microeconomics, but the typical notation there serves to obscure rather than elucidate the fundamental point that partial derivatives can't hold "everything else" fixed.

Anyway, economists frequently look at statistical relationships between variables and seek to tease out causal relationships; if increases in investment tend to be followed by economic growth, for example, one might suppose that the growth is caused by the increase in investment, but it seems logical that investors would invest more when they expect economic growth than when they don't, so even the time-ordering in this case may run reverse to the causal relationship.*  Of course, it's possible that both causal relationships exist, or even that neither does; perhaps both are being driven by something else.  The data do not — indeed, no set of data can — make a distinction between causal explanations.  In order to tease out causal effects econometricians will employ "instrumental variables", but even in that case instruments are useful at implying causation because there is an outside theoretical reason to believe that they bring in a causal relationship themselves; without outside theory, simply throwing more variables at the problem can not give causation.

At this point, then, it is worth asking what causality means.  Perhaps something causes us to expect an increase in investment that isn't connected to other changes we usually expect to affect GDP, so we predict an increase in GDP, or perhaps there's an increase in GDP forecasts that is not connected with other things we expect to drive investment, so we expect an increase in investment.  Presumably this means we believe something is happening that wasn't happening before — something we expect to change the data-generating process.  From that standpoint, causal interpretations of empirical data affect out-of-sample forecasts based on previous observations.

Colloquially, causality is often connected with intention.  If increases in investment cause later increases in GDP, perhaps we can increase investment, and that will increase GDP.  If this is the case, perhaps it matters how "we" — presumably action by the government or some other relatively unitary actor — go about increasing investment.  Formally, perhaps there is a set of choice variables available to "us", and the DGP is presumed to be a function of those choice variables; here, again, if we're looking to ascertain the distribution of variables effected by a set of choices for which we have no empirical data from empirical data we do have (for other values of our choice variables), this is essentially again out-of-sample prediction, and the extrapolation is again going to have to be theory-driven.

So this is how I'm currently thinking about causality, at least from an econometric point of view: Causality is simply about the ways in which I think an underlying data generating process is likely to change, and those ways in which it is not.  (I may think differently next month.)

* Indeed, Le Chatlier noted that near a locally stable equilibrium, effects tend to inhibit their causes; if both take place with a lag, you may see a diagonal time/variable cross-correlation of either sign at various lags, as one causal relationship outweighs the other.

Friday, May 18, 2012

Game Theory

This post will probably contain no original ideas, though I am rephrasing some ideas I'm coming across.

Van Damme (1989)

There are a continuum of Nash equilibria of this game in which player 1 plays T, in all of which player 2 plays C with positive probability; there are no equilibria in which player 1 plays D. With common knowledge of (near) rationality, then, it would seem that player 1 should never be expected to play D; if the possibility of D is excluded, then C is weakly dominated. Van Damme thus argues that (U,L) is the only "reasonable" equilibrium.

It occurs to me that the other example I wanted to present requires extensive form, and I don't feel like drawing game trees.  I'll note that Mertens (1995) in Games and Economic Behavior 8:378–388 constructed two game trees with unique subgame perfect equilibria and the same reduced normal form such that the equilibria of the two games are different.  Elmes and Reny had a 1994 JET paper improving on a classic result that claims to prove that any two extensive form games with the same reduced normal form can be converted to each other by means of three "inessential" transformations that should leave the strategic considerations unaffected.  I have an inclination at some point to reconcile these results for myself.

While I'm here, I will write down one more normal form, not entirely unrelated to the one above:
When deciding between L and C, player 2 can decide as though player 1 were never playing D; if player 1 does play D, the choice between L and C doesn't matter. Similarly, player 1 can choose between U and M as though player 2 is definitely playing L or C. In the game in which choices R and D are disallowed, there's obviously a single Nash equilibrium; thus we should reasonably expect that player 1 will only play U and M with equal probability, and player 2 will only play L and C with equal probability. The game then reduces to
which has a single equilibrium in undominated strategies.

Thursday, May 10, 2012

Lindahl pricing and collective action problems

A "public good" in economics is, as the definition has it, non-rival and non-excludable.  Ignore the "excludable" bit; the idea of non-rivalry is that certain economic products -- a park, perhaps, or an uncongested road -- can be useful to me even if you're using it at the same time.  Information is the only example I can think of of a very cleanly non-rival good; as is contained in the qualifier "uncongested", a lot of goods are "non-rival" in certain situations, but might take on a more rival character in others.  Still, a twenty acre park might be worth more to me than a ten acre park, even if the ten acre park isn't at all crowded, and a lot of goods are essentially non-rival a lot of the time.

The question, then, becomes how to pay for them, and how much of them to produce.  If you want to eat an apple, since that prevents anyone else from eating the apple, it's straightforward to say that whoever eats the apple pays for it, and that the number of apples produced is the number for which people are willing to pay the costs, but for non-rival goods, charging everyone the same amount is likely to be inefficient if different people have different values on the good.  What the Swedish economist Lindahl pointed out, though, is that if you know, for example, that one person values parkland three times as much as another person, and you make the first person pay three times as large a share of the cost of the park, they will agree on how large to make the park; if the relevant parkland is going to cost $5000 per acre, of which Alice pays $15 and Bob pays $5, then if 15 acres is the point at which an extra acre is worth an extra $5 to Bob, it is also the point at which it is worth an extra $15 to Alice, and that is the point where they both say, "make it this large, and no larger."  Further, if the other $4,980 per acre is being raised in the same manner, everyone else agrees; 15 acres is in fact Pareto efficient at this point.

On some level, though, we've simply moved the free-rider problem; if you directly ask your subjects how much they value extra parkland, they have an incentive to low-ball the estimate, such that they can enjoy the benefit of the new park without paying as much for it themselves.  If the municipal authority knew how much Alice and Bob valued park in the first place, they wouldn't have to ask how much park to build; perhaps it's easier to estimate a ratio, but there is still an information problem that's central to this situation.  Some clever theorists have constructed means of, essentially, asking Bob how much Alice values parkland, and asking Alice how much Bob values it; in some situations, the citizens might know more about each others' values than the city manager, but these mechanisms tend to be incomplete at best and usually fatally impractical.

Now, a Lindahl mechanism at least does create some incentive for stating a higher value for the use of the park than if we simply build it such that we can raise voluntary donations for it; if I state a higher value, it makes it more likely that other people will go along with voting to increase the size of the park.  In particular, suppose that 40% of the neighborhood has a particular fondness for parkland and has solved its own collective action problems; if the 40% announce, "we're willing to pay twice as much per person (collectively) as everyone else" before there's a vote on the size of the park, then the other 60% are more likely to support a larger park, and everyone wins.  The magic step there is that the 40% has to solve its own problem first.  I'm trying to figure out whether there's a compelling way to apply the same sort of solution to that problem: for 20% of the population to say to another 20%, "we'll publicly agree to pay 2.5 times our pro-rata share if you'll agree to pay 1.5 times your pro-rata share", and cascade down to the level of the family or tight social groups that might have internal dynamics for solving collective action problems.  It feels like there might be something useful here.