My wife is currently looking for a job; inter alia, this involves having "references", viz. people willing to discuss her with potential employers. This takes time (if nothing else) from the people serving as references, and does not directly benefit them; this is the sort of situation in which some sort of compensation is typically paid from the beneficiary to the benefactor, but if that is done in the case of job references, the payment is non-pecuniary; it follows more of a reciprocal-favors paradigm.
One possible reason for this that just occurred to me this afternoon is that perhaps it is easier to agree on terms-of-trade in a reciprocal-favors paradigm than for the usual unit of account. In bilateral monopoly situations, the possibility of costly bargaining creates an incentive to find default terms on which to focus; for many favors, it might be difficult to figure out a default dollar-price, and therefore costly to bargain to an agreed-upon dollar price for the favor, whereas different favors may seem apparently comparable; even if it's not clear, in dollars, how much each should be worth, it seems that the amount should be about the same. The expense of bargaining is therefore considerably reduced by (implicitly) agreeing on exchange of favors than trying to figure out how much money (or, for that matter, how many goats) a favor is worth.
Thursday, May 2, 2013
Monday, March 25, 2013
fixed exchange rate regimes
I've been taking a history class that has involved reading about the classical gold standard and the various gold exchange standards that prevailed in the western world for about a century. They all involve the idea that a country should peg the value of its currency to something else by promising to buy or sell that something else for that currency, possibly subject to additional stipulations. These days, there are many currencies that are still pegged to other currencies, but I'm not aware of any currency that is actually pegged to something else; the relative prices of various currencies are regulated, but there's nothing external to the system of international currencies to which they are tied either directly or indirectly (for an appropriate narrow conception of "indirectly tied" here).
Even the most comprehensive guarantee of convertibility always included some stipulations; at the very least, to get the US Treasury to buy or sell gold for dollars, you had to get the dollars or the gold to the right location. This meant that at most times and places there could be some deviation from the official rate. At second order, while, under the gold standard, the exchange rate between dollars and British pounds was fixed (since you could use your dollars to buy gold which you could sell for pounds, or vice versa), there were always short-term deviations that were too small or too transient to pay for the cost of shipping gold across the Atlantic ocean. While the rate couldn't move too far from "mint parity", dollars and pounds weren't perfect substitutes; they were, however, much closer than most pairs of commodities studied in economics.
This relied, of course, on the credibility of the gold standard; when countries' gold stocks started to run low, the currency tended to depreciate a bit, in fear that the country, even if it wanted to, would no longer be able to sell gold at the official price, as it would run out of gold to sell.
A few years back there was a penny shortage — businesses were having trouble making sure that they always had enough pennies to make change for customers. The classical economic solution to a shortage is an increase in price; in principle, one might have expected at least some businesses to offer 20 cents' credit for 19 pennies, for example. I'm not aware of that having happened; they seem to have limited themselves to persuasion. Even more than in the case of other goods, people have a strong sense of a "just price" for a penny, I think, and resist its being floated. In a perfect market with continuously diminishing marginal utility, the relative marginal value different people attribute to different goods should be the same, so long as each has some that they could trade; if they didn't, the relative market price of the goods would allow at least one of the two people to trade the relatively expensive object for more of the relatively cheap object to get more of what they value than they give up. It very much seems in this case as though the businesses' relative values for pennies exceeded their market prices, but, with fixed prices, they ran into quantity constraints. If the businesses could have purchased 50 pennies for 50 cents from banks, of course, they would have done so in such circumstances; however, banks, too, including federal reserve banks, were running into quantity constraints as well. The federal reserve system was, for a time, unable to defend the mint parity of the penny; I imagine that the expectation that they would ultimately be able to do so is the only reason the price stayed approximately at mint parity, and custom took over from there to keep it exact.
This idea of different denominations of physical currency representing different mediums of exchange that are held in a fixed relative exchange rate presumes that each penny (for example) is itself a perfect substitute for any other penny. There have been historical cases in which different coins of nominally the same denomination were not treated as such, and indeed it was my very recent experience with this phenomenon that prompted this post. Yesterday, in the Knoxville airport, I was given change that included a $5 bill that was stapled together; today I used that to pay for lunch. The guy who received it expressed aloud his displeasure with it; I myself wasn't particularly pleased to receive it, but didn't try to do anything about beyond pass it along in my next cash purchase. The US Treasury regards it as identical to any other $5 bill, but I imagine it will change hands rather more rapidly than the average $5 bill does, at least until it goes back to the US Treasury's shredders.
Even the most comprehensive guarantee of convertibility always included some stipulations; at the very least, to get the US Treasury to buy or sell gold for dollars, you had to get the dollars or the gold to the right location. This meant that at most times and places there could be some deviation from the official rate. At second order, while, under the gold standard, the exchange rate between dollars and British pounds was fixed (since you could use your dollars to buy gold which you could sell for pounds, or vice versa), there were always short-term deviations that were too small or too transient to pay for the cost of shipping gold across the Atlantic ocean. While the rate couldn't move too far from "mint parity", dollars and pounds weren't perfect substitutes; they were, however, much closer than most pairs of commodities studied in economics.
This relied, of course, on the credibility of the gold standard; when countries' gold stocks started to run low, the currency tended to depreciate a bit, in fear that the country, even if it wanted to, would no longer be able to sell gold at the official price, as it would run out of gold to sell.
A few years back there was a penny shortage — businesses were having trouble making sure that they always had enough pennies to make change for customers. The classical economic solution to a shortage is an increase in price; in principle, one might have expected at least some businesses to offer 20 cents' credit for 19 pennies, for example. I'm not aware of that having happened; they seem to have limited themselves to persuasion. Even more than in the case of other goods, people have a strong sense of a "just price" for a penny, I think, and resist its being floated. In a perfect market with continuously diminishing marginal utility, the relative marginal value different people attribute to different goods should be the same, so long as each has some that they could trade; if they didn't, the relative market price of the goods would allow at least one of the two people to trade the relatively expensive object for more of the relatively cheap object to get more of what they value than they give up. It very much seems in this case as though the businesses' relative values for pennies exceeded their market prices, but, with fixed prices, they ran into quantity constraints. If the businesses could have purchased 50 pennies for 50 cents from banks, of course, they would have done so in such circumstances; however, banks, too, including federal reserve banks, were running into quantity constraints as well. The federal reserve system was, for a time, unable to defend the mint parity of the penny; I imagine that the expectation that they would ultimately be able to do so is the only reason the price stayed approximately at mint parity, and custom took over from there to keep it exact.
This idea of different denominations of physical currency representing different mediums of exchange that are held in a fixed relative exchange rate presumes that each penny (for example) is itself a perfect substitute for any other penny. There have been historical cases in which different coins of nominally the same denomination were not treated as such, and indeed it was my very recent experience with this phenomenon that prompted this post. Yesterday, in the Knoxville airport, I was given change that included a $5 bill that was stapled together; today I used that to pay for lunch. The guy who received it expressed aloud his displeasure with it; I myself wasn't particularly pleased to receive it, but didn't try to do anything about beyond pass it along in my next cash purchase. The US Treasury regards it as identical to any other $5 bill, but I imagine it will change hands rather more rapidly than the average $5 bill does, at least until it goes back to the US Treasury's shredders.
Wednesday, February 20, 2013
non-Smith winner of approval voting equilibrium
Suppose I have 3 voters voting for 3 candidate outcomes. Their preferences are A>C>B, B>C>A, and, since C stands for Condorcet, voter 3's first choice is C. If preferences are strict and common knowledge, a voting scheme in which voters vote first between C and not C and then, in the event of not C, vote between A and B, will result in C's winning the first round, regardless of how you fill out 3's preferences. If voter 3 is exactly indifferent between the bottom two choices, and 1 and 2 are approximately (in a von Neumann-Morgenstern sense) indifferent between their bottom two choices, then "not C" wins the first round; each of the first two voters is willing to take the chance that 3's coin comes up their way. (They are hoping for opposite outcomes, obviously).
Now, in a post about half a year ago, I conjectured that, when there is a Condorcet winner, that candidate will (almost always) win in any approval voting equilibrium in the environment in which I generally think about voting systems: a lot of voters, maximizing expected utility, who know enough about everyone else to predict the vote outcomes with high relative precision but low absolute precision (i.e. if a candidate is getting about 1,000,000 votes, voters' best guesses will be off by more than 10 votes but less than 10,000 votes). The first paragraph scales up to a counter-example to that conjecture; if you have 1,000,000 voters, with about one third of them of each of the preferences given, and each voter believes that A and B have equal chances of winning but that C has a negligible chance of winning, then A and B each get (about) 500,000 votes and C gets 333,000. There's another equilibrium in which the three practically tie, and each wins with a 1/3 probability.[1] There's also an equilibrium that lies "in between" them, where C has just enough chance of winning to induce most C voters not to vote for their second choice, but a low enough chance that a few of them will vote for A and B, which thereby get slightly more votes than C; this equilibrium is unstable in the sense that a slightly different belief would lead to one of the other equilibrium outcomes instead. Both of the other equilibria are robust in this sense.
From a normative standpoint, I have to say, I don't think these are bad equilibria, especially ex ante. I probably want to think more about this in light of some of the nice properties of Condorcet winners, many of which properties are described in terms that suggest that preference order is the only thing about preference that matters[2] — which suggests, perhaps, that the Condorcet solution concept is best ex post, but that something like a continuous approval vote might give better results ex ante in some contexts.
[1] In both cases the marginal probability of winning is practically equal to the probability conditional on there being a close race, which is what really matters, since they're both basically tied. I use the term "probability" here for conciseness.
[2] Note that the von Neumann-Morganstern utilities in the example given are essential; if the voters prefer their second choices to a coin toss between the other two — say, for definiteness, that half of the C voters prefer A to B and the other half prefer B to A — the construction in the first paragraph fails, and I'm pretty sure that now the only equilibrium is in fact the one in which every voter votes for two choices, giving C 1,000,000 votes and A and B 500,000 each.
Now, in a post about half a year ago, I conjectured that, when there is a Condorcet winner, that candidate will (almost always) win in any approval voting equilibrium in the environment in which I generally think about voting systems: a lot of voters, maximizing expected utility, who know enough about everyone else to predict the vote outcomes with high relative precision but low absolute precision (i.e. if a candidate is getting about 1,000,000 votes, voters' best guesses will be off by more than 10 votes but less than 10,000 votes). The first paragraph scales up to a counter-example to that conjecture; if you have 1,000,000 voters, with about one third of them of each of the preferences given, and each voter believes that A and B have equal chances of winning but that C has a negligible chance of winning, then A and B each get (about) 500,000 votes and C gets 333,000. There's another equilibrium in which the three practically tie, and each wins with a 1/3 probability.[1] There's also an equilibrium that lies "in between" them, where C has just enough chance of winning to induce most C voters not to vote for their second choice, but a low enough chance that a few of them will vote for A and B, which thereby get slightly more votes than C; this equilibrium is unstable in the sense that a slightly different belief would lead to one of the other equilibrium outcomes instead. Both of the other equilibria are robust in this sense.
From a normative standpoint, I have to say, I don't think these are bad equilibria, especially ex ante. I probably want to think more about this in light of some of the nice properties of Condorcet winners, many of which properties are described in terms that suggest that preference order is the only thing about preference that matters[2] — which suggests, perhaps, that the Condorcet solution concept is best ex post, but that something like a continuous approval vote might give better results ex ante in some contexts.
[1] In both cases the marginal probability of winning is practically equal to the probability conditional on there being a close race, which is what really matters, since they're both basically tied. I use the term "probability" here for conciseness.
[2] Note that the von Neumann-Morganstern utilities in the example given are essential; if the voters prefer their second choices to a coin toss between the other two — say, for definiteness, that half of the C voters prefer A to B and the other half prefer B to A — the construction in the first paragraph fails, and I'm pretty sure that now the only equilibrium is in fact the one in which every voter votes for two choices, giving C 1,000,000 votes and A and B 500,000 each.
Friday, February 1, 2013
random allocation mechanisms
I've been thinking again about a classic allocation problem: we have n agents, each of which has preferences over n items, and we wish to assign one of the items to each agent. An allocation is ex post efficient if and only if it can result from random serial dictatorship, which is probably the way you've solved any such problem you've encountered in real life: the people draw straws (at some level of metaphorical remove), and one person gets his first choice, the next gets his first choice among those remaining, etc. It is fairly obvious that this will result in an allocation that is Pareto efficient (as long as agents have strict preferences), i.e. that there is no trade available after the fact that would make both parties to the trade better off (whichever agent went before the other would not want to trade); it is only slightly less involved to show that any Pareto efficient outcome can be generated by giving the agents the opportunity to pick in some order. (Lemma: at least one agent will get its first choice, or else a mutually beneficial trade would in fact exist. Make that person the first one to choose, then consider the remaining assignment problem with n-1 agents and n-1 items. The result follows from induction.)
What's left to say? Well, it turns out that there are situations in which random serial dictatorship is not "ex ante" efficient: once the mechanism has run, there is always one person who would object if you said, "hey, let's do this assignment differently," but if the people are mathematically adept and know each others' preferences, there are situations in which all agents would prefer some other mechanism from the get-go. From Bogomolnaia and Moulin's 2001 paper in JET, with items a, b, c, and d, suppose two agents prefer a>b>c>d and the other two agents prefer b>a>d>c. If the first two agents get a and c (they toss a coin to decide who gets which), and the other two get b and d, then each agent has a 1/2 chance of getting its top choice and will always get at least its third choice, while random serial dictatorship gives each agent only a 5/12 chance of getting its top choice and 11/12 chance of getting at least its third choice (i.e. a 1/12 chance of getting its last choice). An AER paper from 2011 (I think) offers an even simpler example if one is willing to use von-Neumann–Morgenstern utilities instead of the (stronger) concept of first order stochastic dominance: given 3 items, if three agents assign u(A)=3 and u(B)=0 but two of them assign u(C)=1 and one assigns u(C)=2, random serial dictatorship gives each agent each item with a probability of 1/3, but all agents get a higher ex ante expected utility from giving item C to the third agent and letting the other two toss a coin for A and B.
Generically, suppose I have expected utility maximizing agents, and I want to investigate what kinds of mechanisms I could use to assign them (randomly) objects in this fashion. If I know the agents' utility functions, and I'm a benevolent mechanism designer, I can calculate Pareto optimal mixtures of assignments. It turns out that (I'm reasonably sure) the resulting assignment can always be expressed in the following fashion: there is some positive affine transformation applied to the utility of each agent (possibly different transformations for different agents) such that the ex post utility obtained by each agent will be at least as high as the utility any other agent would have obtained from the object that agent is assigned. For example, if there is a positive probability that you will get item A, then your utility (after the necessary affine transformation) for item A is at least as high as anyone else's utility (after the necessary affine transformation) for item A. In the example of the previous paragraph, in fact, I don't have to do any affine transformations from the form in which I gave the utility functions; the agent who ends up with A assigns it utility 3, the agent who ends up with B assigns it 0, and the agent who ends up with C assigns it 2, regardless of how the coin toss turns out.
Usually, however, I want to work in an environment in which I, as the mechanism designer, don't know their utilities, so my mechanism will have to elicit their preferences in some fashion, and assign random outcomes with a probability that depends on what they indicate. Whatever mechanism I use, however weird, it will ultimately result in some mapping from sets of utility functions to some probability distribution over assignments. These probability distributions then generate expected utilities for each agent. In considering all possible mechanisms, it becomes convenient to ignore the details of the mechanisms (at least for a while), and simply consider the properties of the induced map that takes a utility function from each agent and produces an expected utility for each agent. If the agents know how the mechanism works, though, they may lie; I can only use mechanisms where no agent would prefer the outcome he could obtain by claiming to have a different utility than what he really does. If I consider the mechanisms that are "incentive compatible", and just consider the problem from the standpoint of a single agent, given whatever that agent knows about other agents' actions, the expected utility as a function of what that agent does must be "incentive compatible". It turns out that this imposes relatively few constraints, and they include a constraint we might prefer to have anyway: the expected utility must transform under positive affine transformations in the same way as the utility function does. That is to say, if reporting "u" gets you an expected utility of 5, then reporting "2u+3" must get you 13. Both functions represent the same preferences, and it is necessary that they give the same actual outcome, however it's parameterized. The only other constraint that incentive compatibility imposes is convexity. Since the function is necessarily homogeneous of degree 1 and is convex, it is the support function of some set; that is to say, whatever the mechanism is, given other agent's actions (to the extent this agent knows them), there is some set of lotteries over assignments such that this agent's expected utility from the mechanism will be the expected utility that the agent would derive from its favorite element of the set. [Update: this is actually obvious for a simpler reason. Consider the set of lotteries consisting of any lottery that would be assigned to some utility function; then if the agent, for any utility function, is assigned a lottery to which he prefers some other element of the set, that agent will lie, claiming to have the utility of an agent who is assigned the element he prefers.]
This is appealing in some ways, though unappealing in others. In a large n environment with symmetry among the agents, I might imagine that I can announce the effective set of lotteries, and that each agent is choosing among the same such set. Now, in fact, the set of options an agent has will generally depend on the choices other agents make; the announcement, therefore, would have to be made after everyone's action is taken, so this "announcement" would not be "here's your choice of lotteries, which do you prefer?", since I need your action in order to find out other agents' sets of options. It could be a useful good-faith auditing mechanism, though, to say "this is the set of options that were generated, and thus this is the lottery you chose by way of your action that implied that it was your preference, and the result of the lottery is that you get object H". However, while the effective set of choices will generally depend on other agents' actions, incentive compatibility requires that the set cannot depend on this given agent's action; in a large n environment with a normal amount of information, it might be that we can produce a set that depends only slightly on each agent's action, and thus is practically incentive compatible. It is not, however, strictly incentive compatible.
Suppose we're looking for "Nash equilibrium" type settings, in which everyone knows what everyone else will do but still chooses to do what they were going to do anyway, and we want to spell out a full mechanism. The actions of every set of n-1 agents determine the choices available to the final agent, who is free to choose among them; on the other hand, if the agent gets a real choice, the other n-1 agents will have to, by their actions, be selecting "whatever's left". "Nash equilibrium" type settings, though, may not make practical sense; the mechanism design literature includes implementation in Nash equilibrium by taking advantage of the idea that everyone knows everyone else's preferences, so that you can ask agents for each other's information. (I've assumed this away, implicitly, by suggesting that one's action is only a function of one's own preferences.)
It seems likely to me that, in situations with large numbers of individuals, I can generate shadow prices for the options such that each agent, in a pretty good Pareto efficient equilibrium, has (at least approximately) its preferred lottery among those with expected shadow price of 0 or less. I'm not quite sure of that, though. In a particular small case, suppose three individuals have preferences B>C>A, A>C>B, and A>B>C, with each indifferent between its second choice and a coin toss between the first and third choices; a coin toss between allocations BCA and CAB is, I believe, Pareto efficient, and seems in some ways like the logical lottery to hold, but it can't fit the preceding description, since 50/50 lotteries between each pair must be part of the choice set (and thus have non-positive expected shadow price) but no single item may be part of the choice set (and thus presumably each of them has a positive shadow price). Perhaps the next questions for me to answer would be 1) is there a different Pareto efficient outcome here that can be generated in the proposed fashion, 2) are these indeed part of an incentive compatible mechanism, and 3) can I characterize under what conditions this becomes a good scheme?
Update: Oh, well, I suppose I've well established that the choice sets depend on what the other agents are doing, so all that's really necessary here is that agent 1 have a 50/50 B/C option, agent 2 have a 50/50 A/C option, and agent 3 having a 50/50 A/B option, all as the respective best options. B, A, and A must respectively not be available in higher probabilities with these mixtures, so if agent 1 sees shadow prices of B>0>C, agent 2 sees A>0>C, and agent 3 sees A>0>B with equal absolute values, we could get choice sets of these sorts. In particular, B has a positive shadow price for agent 1 and a negative shadow price for agent 3; agents 1 and 2 together in some sense overdemand B, while agents 2 and 3 in some sense underdemand it. It seems reasonable to think that any option that is the first choice of one agent would have a positive shadow price for other agents in a reasonable symmetric mechanism. More generally, if there are m other agents whose first m choices are the same, then they and I can't all be allowed to choose a lottery that guarantees us an element of that set, so it seems likely that in a symmetric mechanism that, any time m agents' top m choices are the same, all other agents face a positive shadow price for each of the m items. Note, though, that the "Boston mechanism" violates this principle; if two agents have A as their first choice and B as their second, the Boston mechanism would allow other agents to simply take choice B.
Update: Suppose agents are restricted to a set U of possible utility vectors such that 1) if two elements of U are distinct, they indicate different preferences -- i.e. if u is in U, then 2*u+3 is not, and 2) U is convex. Now consider the following program: On the space Un×Rn find the graph of the correspondence from utility function profiles to feasible expected utility profiles, find a way to define and construct the closure of the convex hull of the upper contour set, and hope that its boundary represents the graph of a function from Un→Rn. That thing should be convex, and "optimal" in some sense. Can it be constructed intelligibly, and, if so, can it be done in a computationally efficient manner?
I suppose another angle is in fact to start from serial random dictatorship as a benchmark; what mechanisms might pareto-dominate it, or (in some sense) nearly do so? In the special case where the other agents all have the same preferences (preferring 1>2>3>...>n), I can choose a lottery that gives me each item with probability 1/n, or I can shift some probability from more preferred to less preferred (by other agents) items, but the constraint I face can't be fully "linearized"; for n=3, I can go from (1/3,1/3,1/3) to (0,2/3,1/3), but I can't go to (0,0.67,0.33), which I might imagine if I'm thinking in terms of trading 100/300 of the most popular item and 1/300 of the least popular in exchange for 101/300 of the second-most-popular; allowing this sort of exchange is precisely where I might expect to be able to offer gains from trade between agents with the same ordering but different vN-M preferences.
What's left to say? Well, it turns out that there are situations in which random serial dictatorship is not "ex ante" efficient: once the mechanism has run, there is always one person who would object if you said, "hey, let's do this assignment differently," but if the people are mathematically adept and know each others' preferences, there are situations in which all agents would prefer some other mechanism from the get-go. From Bogomolnaia and Moulin's 2001 paper in JET, with items a, b, c, and d, suppose two agents prefer a>b>c>d and the other two agents prefer b>a>d>c. If the first two agents get a and c (they toss a coin to decide who gets which), and the other two get b and d, then each agent has a 1/2 chance of getting its top choice and will always get at least its third choice, while random serial dictatorship gives each agent only a 5/12 chance of getting its top choice and 11/12 chance of getting at least its third choice (i.e. a 1/12 chance of getting its last choice). An AER paper from 2011 (I think) offers an even simpler example if one is willing to use von-Neumann–Morgenstern utilities instead of the (stronger) concept of first order stochastic dominance: given 3 items, if three agents assign u(A)=3 and u(B)=0 but two of them assign u(C)=1 and one assigns u(C)=2, random serial dictatorship gives each agent each item with a probability of 1/3, but all agents get a higher ex ante expected utility from giving item C to the third agent and letting the other two toss a coin for A and B.
Generically, suppose I have expected utility maximizing agents, and I want to investigate what kinds of mechanisms I could use to assign them (randomly) objects in this fashion. If I know the agents' utility functions, and I'm a benevolent mechanism designer, I can calculate Pareto optimal mixtures of assignments. It turns out that (I'm reasonably sure) the resulting assignment can always be expressed in the following fashion: there is some positive affine transformation applied to the utility of each agent (possibly different transformations for different agents) such that the ex post utility obtained by each agent will be at least as high as the utility any other agent would have obtained from the object that agent is assigned. For example, if there is a positive probability that you will get item A, then your utility (after the necessary affine transformation) for item A is at least as high as anyone else's utility (after the necessary affine transformation) for item A. In the example of the previous paragraph, in fact, I don't have to do any affine transformations from the form in which I gave the utility functions; the agent who ends up with A assigns it utility 3, the agent who ends up with B assigns it 0, and the agent who ends up with C assigns it 2, regardless of how the coin toss turns out.
Usually, however, I want to work in an environment in which I, as the mechanism designer, don't know their utilities, so my mechanism will have to elicit their preferences in some fashion, and assign random outcomes with a probability that depends on what they indicate. Whatever mechanism I use, however weird, it will ultimately result in some mapping from sets of utility functions to some probability distribution over assignments. These probability distributions then generate expected utilities for each agent. In considering all possible mechanisms, it becomes convenient to ignore the details of the mechanisms (at least for a while), and simply consider the properties of the induced map that takes a utility function from each agent and produces an expected utility for each agent. If the agents know how the mechanism works, though, they may lie; I can only use mechanisms where no agent would prefer the outcome he could obtain by claiming to have a different utility than what he really does. If I consider the mechanisms that are "incentive compatible", and just consider the problem from the standpoint of a single agent, given whatever that agent knows about other agents' actions, the expected utility as a function of what that agent does must be "incentive compatible". It turns out that this imposes relatively few constraints, and they include a constraint we might prefer to have anyway: the expected utility must transform under positive affine transformations in the same way as the utility function does. That is to say, if reporting "u" gets you an expected utility of 5, then reporting "2u+3" must get you 13. Both functions represent the same preferences, and it is necessary that they give the same actual outcome, however it's parameterized. The only other constraint that incentive compatibility imposes is convexity. Since the function is necessarily homogeneous of degree 1 and is convex, it is the support function of some set; that is to say, whatever the mechanism is, given other agent's actions (to the extent this agent knows them), there is some set of lotteries over assignments such that this agent's expected utility from the mechanism will be the expected utility that the agent would derive from its favorite element of the set. [Update: this is actually obvious for a simpler reason. Consider the set of lotteries consisting of any lottery that would be assigned to some utility function; then if the agent, for any utility function, is assigned a lottery to which he prefers some other element of the set, that agent will lie, claiming to have the utility of an agent who is assigned the element he prefers.]
This is appealing in some ways, though unappealing in others. In a large n environment with symmetry among the agents, I might imagine that I can announce the effective set of lotteries, and that each agent is choosing among the same such set. Now, in fact, the set of options an agent has will generally depend on the choices other agents make; the announcement, therefore, would have to be made after everyone's action is taken, so this "announcement" would not be "here's your choice of lotteries, which do you prefer?", since I need your action in order to find out other agents' sets of options. It could be a useful good-faith auditing mechanism, though, to say "this is the set of options that were generated, and thus this is the lottery you chose by way of your action that implied that it was your preference, and the result of the lottery is that you get object H". However, while the effective set of choices will generally depend on other agents' actions, incentive compatibility requires that the set cannot depend on this given agent's action; in a large n environment with a normal amount of information, it might be that we can produce a set that depends only slightly on each agent's action, and thus is practically incentive compatible. It is not, however, strictly incentive compatible.
Suppose we're looking for "Nash equilibrium" type settings, in which everyone knows what everyone else will do but still chooses to do what they were going to do anyway, and we want to spell out a full mechanism. The actions of every set of n-1 agents determine the choices available to the final agent, who is free to choose among them; on the other hand, if the agent gets a real choice, the other n-1 agents will have to, by their actions, be selecting "whatever's left". "Nash equilibrium" type settings, though, may not make practical sense; the mechanism design literature includes implementation in Nash equilibrium by taking advantage of the idea that everyone knows everyone else's preferences, so that you can ask agents for each other's information. (I've assumed this away, implicitly, by suggesting that one's action is only a function of one's own preferences.)
It seems likely to me that, in situations with large numbers of individuals, I can generate shadow prices for the options such that each agent, in a pretty good Pareto efficient equilibrium, has (at least approximately) its preferred lottery among those with expected shadow price of 0 or less. I'm not quite sure of that, though. In a particular small case, suppose three individuals have preferences B>C>A, A>C>B, and A>B>C, with each indifferent between its second choice and a coin toss between the first and third choices; a coin toss between allocations BCA and CAB is, I believe, Pareto efficient, and seems in some ways like the logical lottery to hold, but it can't fit the preceding description, since 50/50 lotteries between each pair must be part of the choice set (and thus have non-positive expected shadow price) but no single item may be part of the choice set (and thus presumably each of them has a positive shadow price). Perhaps the next questions for me to answer would be 1) is there a different Pareto efficient outcome here that can be generated in the proposed fashion, 2) are these indeed part of an incentive compatible mechanism, and 3) can I characterize under what conditions this becomes a good scheme?
Update: Oh, well, I suppose I've well established that the choice sets depend on what the other agents are doing, so all that's really necessary here is that agent 1 have a 50/50 B/C option, agent 2 have a 50/50 A/C option, and agent 3 having a 50/50 A/B option, all as the respective best options. B, A, and A must respectively not be available in higher probabilities with these mixtures, so if agent 1 sees shadow prices of B>0>C, agent 2 sees A>0>C, and agent 3 sees A>0>B with equal absolute values, we could get choice sets of these sorts. In particular, B has a positive shadow price for agent 1 and a negative shadow price for agent 3; agents 1 and 2 together in some sense overdemand B, while agents 2 and 3 in some sense underdemand it. It seems reasonable to think that any option that is the first choice of one agent would have a positive shadow price for other agents in a reasonable symmetric mechanism. More generally, if there are m other agents whose first m choices are the same, then they and I can't all be allowed to choose a lottery that guarantees us an element of that set, so it seems likely that in a symmetric mechanism that, any time m agents' top m choices are the same, all other agents face a positive shadow price for each of the m items. Note, though, that the "Boston mechanism" violates this principle; if two agents have A as their first choice and B as their second, the Boston mechanism would allow other agents to simply take choice B.
Update: Suppose agents are restricted to a set U of possible utility vectors such that 1) if two elements of U are distinct, they indicate different preferences -- i.e. if u is in U, then 2*u+3 is not, and 2) U is convex. Now consider the following program: On the space Un×Rn find the graph of the correspondence from utility function profiles to feasible expected utility profiles, find a way to define and construct the closure of the convex hull of the upper contour set, and hope that its boundary represents the graph of a function from Un→Rn. That thing should be convex, and "optimal" in some sense. Can it be constructed intelligibly, and, if so, can it be done in a computationally efficient manner?
I suppose another angle is in fact to start from serial random dictatorship as a benchmark; what mechanisms might pareto-dominate it, or (in some sense) nearly do so? In the special case where the other agents all have the same preferences (preferring 1>2>3>...>n), I can choose a lottery that gives me each item with probability 1/n, or I can shift some probability from more preferred to less preferred (by other agents) items, but the constraint I face can't be fully "linearized"; for n=3, I can go from (1/3,1/3,1/3) to (0,2/3,1/3), but I can't go to (0,0.67,0.33), which I might imagine if I'm thinking in terms of trading 100/300 of the most popular item and 1/300 of the least popular in exchange for 101/300 of the second-most-popular; allowing this sort of exchange is precisely where I might expect to be able to offer gains from trade between agents with the same ordering but different vN-M preferences.
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.
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
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
- 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
- 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.
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
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
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,
which might lead to something like
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):
| Notre Dame | |||
| Oregon | Florida | Oregon | |
| Georgia | Texas A&M | Texas A&M | |
| Texas A&M | Oregon | ||
| Alabama | Alabama | ||
| Florida | Notre Dame | Alabama | |
| LSU | Stanford | Notre Dame | |
| Stanford |
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
| a | A4-N5 | Wa-Lb |
| b | A1-N2 | Wb-Wc |
| c | N1-A2 | |
| d | N4-A5 | Wd-Lc |
| e | A3-N6 | We-Wf |
| f | N3-A6 |
| AFC 1 | w | (loser) | w |
| NFC 2 | |||
| NFC 1 | w | ||
| AFC 2 | |||
| AFC 3 | w | w | |
| NFC 6 | |||
| NFC 3 | w | ||
| AFC 6 | |||
| AFC 1 | (loser) | w | w |
| NFC 2 | |||
| AFC 4 | w | ||
| NFC 5 | |||
| NFC 1 | (loser) | w | |
| AFC 2 | |||
| NFC 4 | w | ||
| AFC 5 |
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.
| AFC | NFC |
| Denver | Atlanta |
| New England | San Francisco |
| Houston | Green Bay |
| Indianapolis | Seattle |
| Baltimore | Washington |
| Cincinnati | Minnesota |
| Denver | San Francisco | (New England) | New England |
| San Francisco | |||
| Atlanta | New England | ||
| New England | |||
| Houston | Houston | Green Bay | |
| Minnesota | |||
| Green Bay | Green Bay | ||
| Cincinnati | |||
| Denver | (Denver) | Denver | Baltimore |
| San Francisco | |||
| Indianapolis | Washington | ||
| Washington | |||
| Atlanta | (Atlanta) | Baltimore | |
| New England | |||
| Seattle | Baltimore | ||
| Baltimore |
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.
Subscribe to:
Posts (Atom)
