 Hi folks, it's Matt again, and we're going to talk now about dominant strategy implementation So the idea that we're looking at is we have a society we have to make a decision and we're trying to design a mechanism that is going to take people's preferences and give us outcomes and In particular, let's start by looking at a situation where we've got a full set of alternatives and any possible ranking over those things So for instance, we have a set of candidates that we need to pick from Say four or five six candidates. Okay, so we think about the voting rules that we looked at earlier now We've got to make a choice over these candidates and in particular what we'd like is when we ask people that they have no Difficulties in deciding what they should announce. They should just be truthful. Okay, so we want dominant strategy implementation We want to design the mechanism so that they want to tell us truthfully exactly what their preference rankings are over the alternatives And they don't gain at all by trying to mix up the alternatives and manipulate the system. Okay, so that's the idea and In particular, let's think of we've got our society with n individuals Finite set of outcomes. Oh and if we begin to think about designing mechanisms in this world And we want one where every agent has a dominant strategy for each preference we can invoke the revelation principle and the revelation principle tells us that if we do have an indirect mechanism that has dominant strategies, we can just collapse that into a Social choice function a direct mechanism where people just tell us directly their preferences And then we give them the outcomes they would have gotten through the original mechanism for each announcement of their preferences So if they'd followed the strategies that they had had dominant strategies in the original one So this makes truth a dominant strategy. So the revelation principle will means that we can without loss of generality for this exercise look at social choice functions Directly, okay, and so now what we want to do is think about which social choice functions Can a society have which are going to be dominant strategy? truthful in this sense Okay, well, so it's These things are also known as non manipulable strategy proof sometimes they're called straightforward mechanisms or social choice functions and The important result in this area due to Alan Gibbard and Mark Satterthwaite in the early 1970s It's gonna say we're gonna have a really hard time doing this in a setting where people can have any possible ranking over the alternatives So let's have a look at that. This is what's known as the Gibbard Satterthwaite theorem And it's another form of an impossibility theorem similar to what we saw in terms of arrows Theorem and the Mulder Satterthwaite theorem so situations where we have a set of Conditions we'd like to have and the theorem says it's impossible to have this desirable set of conditions all at once So what's the setting here? We've got a social choice function We'd like to have one that's mapping all possible preferences. So people have linear orders they can have any strict ranking over candidates and We are going to look at situations with at least three outcomes. So we have at least three candidates to choose from and We're going to also look at a social choice function, which is on to that means for every possible outcome there is a profile of preferences which gives you that outcome and that condition Can be satisfied quite easily. For instance, if you just required that your social choice function be unanimous So if all individuals prefer the same alternative, we all say we love candidate a then Society should pick candidate a if you put that minimal condition in then indeed The in this domain of preferences C is going to be on to so if we put those conditions in then what does the theorem say? The theorem says that we're going to have the strategy proofness condition truthful reporting of preferences as a dominant strategy for every agent at Every preference profile if and only if C is dictatorial Okay, so again, that means here There is just some particular individual eye for whom the choice function is just always their favorite alternative There the thing that maximizes their preferences In regardless of what anybody else says so we just pick one individual. We just listen to that person Okay Now in terms of the proof of this It's clear that if we assign somebody to be a dictator and don't listen to anybody else That's going to be strategy proof I nobody else can make any difference doesn't matter what they say and the person who is a dictator always wants to be truthful Because they're getting their favorite alternative The converse of this theorem is is much more difficult the part saying that if it's strategy proof then it has to be dictatorial and the this can be proven by Various means there are proofs that relate this back to arrows theorem and and show that that there are similar Conclusions that can get in terms of the basic steps the lemmas that were proven there There's a very elegant proof by Salvador Barbera which Works off of individuals being pivotal in terms of changing their preferences and changing the outcome And showing that you can't have too many individuals being pivotal at once if you are going to satisfy dominant strategies So there's a basic conflict of allowing people to make decisions and having multiple people make decisions at the same time And making sure that everybody cannot manipulate things by by announcing things falsely So we're not going to explicitly offer that proof, but you can find a various versions of this in the literature and We'll leave you to look at that directly What this means is that any social choice function that we write down that we're interested in assuming that we don't want a Dictatorial functions are going to be manipulable in some situations So if we're in a voting setting and we have a set of Candidates that we're looking at and we have to pick among those candidates and any possible ordering of those is possible in people's preferences Regardless of what rule we use we're going to end up having people manipulate that rule in certain situations So that's a very negative result in some sense It's in a damaging result and and it's different from arrows theorem because what this is doing is Really looking at the incentives that people have and saying it's going to be difficult to be get people to be truthful When we were looking at arrows theorem There was nothing set about whether people were being truthful or not the kinds of conclusions that were being reached in arrows theorem We're just saying that some basic independence Conditions and Pareto conditions couldn't be satisfied by a social welfare ordering at the same time But presuming that you could see what people's preferences were as the input and this is saying that if you're going to make a choice It's going to be difficult to get people to reveal their preferences to you in a truthful manner one more thing about this And this is something you can verify yourself Suppose that you only had two alternatives Then it's going to be easy to find a rule Think of majority rule. So is that strategy proof? So imagine that we have just two candidates and you get to vote for which one you would prefer and you always prefer one to the other There there will be a variety actually of strategy proof rules in that setting If C is not on to Then we are ignoring certain outcomes Then you're going to have some more limited conclusions here that things are going to be dictatorial But only on the range of the outcome function if at least if it still has at least three alternatives in it So there's some limitations on this but in terms of really getting around this kind of Negative result there's various ways we can think of doing this One is to use a weaker form of implementation So here we were we were asking for dominant strategies for all individuals at all preferences We could work instead for something like Bayes Nash implementation Bayesian implementation where we ask people just to have it be a best response to to announce truthfully given that everyone else is and That'll open some doors for us That's going to be much more demanding in terms of an equilibrium concept and the beauty of dominant strategies is people don't have to worry about what other people are doing and They don't have to know or have beliefs about what's going on Bayes Nash Structure a Bayesian Nash structure is much more demanding in that sense. We could Relax the assumption that people have arbitrary preferences and look at more structured settings and indeed when we do that We're going to find a much more interesting and positive results In terms of rules that we can apply so more interesting rules I should say I'm not more interesting results the gibberish tethered way theorem is a very interesting result But we'll find more interesting rules that we can work with we've already seen one in fact median voting on a single peak domain So as we mentioned there if you're asked for your preferences You have no reason to try and distort your preferences the median peak wins and the only way to change that is to flip over and Announce something on the other side which would be worse for a given individual you could also take the max of the peaks You could do instead of median voting have maximum voting So whoever Has the maximum peak or the minimum peak or any order statistics? So single peak domains are going to be settings where what we've done is we've limited the set of preferences So not any ordering over candidates is allowed any longer and once you do that then it's easier to design strategy-proof rules Another setting on trade so imagine that you have an indivisible good You have to a buyer and a seller and they might have private values for how much they Value the good and you're trying to get to trade to go on One mechanism that works in that situation is just to fix a price So say of the you can trade the price at at a price of 10 So individuals say at a price of 10 am I willing to trade or not? And now there's a simple decision. You can't influence the price All you can do is influence whether you actually trade or not And so then it's you have an incentive to say truthfully Do you really want to trade or do you not and then trade if both of the agents want to trade? So there you can design a dominant strategy mechanism. There's going to be some inefficiencies associated with that You're not going to have trade occurring in all the circumstances that you'd like But at least you've got dominant strategies and we'll take a closer look at these things We're going to see a whole series of other types of rules The we'll look at the vicaricard grove schemes and other kinds of schemes and when we look and narrow in the kinds of settings So we're looking at with more structure on preferences We'll have a whole series of interesting dominant strategy compatible mechanisms Which are going to be present and we'll take a look at particular kinds of environments where that's going to be possible We'll be next up