 One can argue that the aerovian social welfare setup is too demanding and that is one of the reasons why We got the impossibility result due to arrow. So In the arrows social welfare setup, we are looking at a complete ordering over the alternatives as an outcome of that welfare function now this is this could be very demanding and the the summary of The arrows impossibility result can be said in the following statement that achieving a social ordering Remember that this f of r is nothing but a social ordering It is giving and taking as input all the preference orders over all the individuals or the individual agents and taking and giving up one complete ordering Having this social ordering in a democratic way is impossible. That is what arrows impossibility result is saying So in order to mitigate this Impossibility they had been various attempts and we will discuss about two such attempts In this module and then following modules So the first setup is what is known as a social choice setting instead of an ordering We are going to the the function that we are going to define will be giving out only one alternative and not a complete ordering And the second way of handling this is we will see certain restrictions on the agent preferences So maybe in in practice agents preferences cannot take all possible orderings over all the alternatives and they have certain restrictions certain Regularities and if we can use those regularities appropriately, maybe we can get rid of this kind of impossibility results So this brings us to this setup of social choice function. So the function that will be Will be focusing on in this setup the social choice setup is what is known as the social choice function Notice that the left hand side remains the same the only difference is that now we are only considering strict preferences Let's assume that for now in the social welfare setup. You are looking at all Preferences which also included Indifferences but here we don't allow for differences and we have only strict preferences So if all these agents all these n agents have their strict preferences given to this social choice function This function f will pop up one alternative rather than the whole ordering over that over that set of alternatives So the most representative example could be that of voting or election So let us spend some time in discussing various voting rules because these are quite common and quite well Studied and therefore for our own knowledge. We can actually Know what are the different kinds of voting rules or election rules? So the first class of voting rules that we are going to discuss are called the scoring rules So based on the position on which a specific voter keeps its alternative. So let's say we have Alternatives a pc and so on and the corresponding score that one gives to each of these positions is given by s1 s2 s3 and so on and When we are looking at the Final outcome. We are just taking so maybe for certain other or agent We may have a have a topmost position. You may have a second position and see Similarly in that case the score will be s2 and s3 s1 s2 s3 for those things. So In if there are these two Voters in this context then we would have a total score of s1 plus s2 Similarly, you would have s1 plus s2 and c would have got twice of s3. So that is what this scoring rule means so we Define scores and of course all these si's Should be at least as much as si plus one. So this scores are non-decreasing in nature And there are various and whoever has the highest score that Yeah, that alternative is Selected as the final outcome. So certainly this is a social choice function because we are taking individual's preferences we are only defining certain scores and Finally, we are popping out one particular alternative Breaking ties arbitrarily. So there are certain special cases here the most Popular one or the things that we can the voting rule where we can Associate most of the time is what is known as plurality. You have the topmost Position the topmost score to be equal to one and all other scores to be equal to zero So you just count the topmost vote and this is exactly how we We have our usual elections Look so far the Vidant so far elections So they have the only The voters only cast their most favorite candidate, which you can consider as Giving the score of one for all the other agents. You don't really ask This agents and the ask the voters to reveal them so the veto is is another type of Scoring rule where you have only one alternative which you don't want to Include for all the other Alternatives you have the same score of one Borda is a is named after a French mathematician who and also a social scientist Who has given this? This kind of a score where you start with the M minus if there are M Different alternatives you start with the M M minus one of as the score of the first one and Then M minus two for the second one And so on up to M the final one So the the s of M minus one will be equal to one and the final one will have a score of zero So this is how this This border voting rule is defined Harmonic has this course to be the harmonic series and For key approval you have the first key most preferred alternatives which is having the score of one and for the rest M minus K you have zeros. That's the set of scoring rule based mechanisms. Now, there are Slight modifications of this plurality rule, which is known as the plurality with runoff So there are two phases in this kind of a voting rule in the first phase all the candidates participate and People cast their votes in the usual plurality form and then the top two candidates are selected and then all the voters are asked to vote again and this is and So after the first phase everybody except the top two has been eliminated. So some of the votes Which were cast by the by certain voters of whose favorite candidate has been eliminated They are given another chance to pick Pick one of the candidates from this top two candidates And this is exactly what is done in the French presidential election. It is done using plurality with runoff We'll discuss a couple of more voting rules. The the third voting rule is the Maximee So what what it does is it selects the candidate with largest margin of victory Now what is a margin of victory? This is the number of votes that you need to delete or alter in order to change the current outcome So for instance a was currently being Happening to be the winner What is the minimum number of votes that you need to change in order to make someone else a winner? And that is called the margin of victory in some sense. This is the the the closest Opposing voter opposing candidate whose votes can be altered So if if someone selects a was beating its closest competitor By a margin of let's say a hundred votes and if that hundred votes can could be altered then you can Make the the closest competitor to be the winner So this is something like the minimum number of votes in order to change the outcome And you are trying to find out who has this highest margin So that is the reason why it is called Maximee The the fourth type is the coplan rule So now we can think about a pair of Candidates and look at what happens in the pair wise election. So we remove all the so we have the total Ordering of all the agents over these candidates and We look at pair wise Let's say a and b and remove all the other candidates So whether they are so who is winning in this pair wise election? And if a wins the pair wise election with Against b then we are giving giving a score of one and similarly for all possible pairs So all m-1 different elections pair wise elections for that particular candidate We are looking at who is becoming the winner. So coplan rule is essentially who has the largest score Based on that pair wise elections He that candidate is going to be the winner Now in this case, there is an interesting observation that you one can make Again, Copeland is it's named after Copeland Similarly, there is a idea of a condor say winner. So now what is a condor say winner? you can think of again the pair wise elections and If there exists one candidate one specific Alternative which beats every other candidate in pair wise election Then we are going to call that winner a condor say winner And it is not guaranteed to exist such kind of a condor say winner may not exist And here is one example very simple example with three voters and three candidates so Maybe these three agents have this three Preferences orders over these three candidates a b and c What you can see is that a is beating B in a pair wise election So a is above b in two cases and b is above a in only one case. So a is beating a in a pair wise election Similarly, see So sorry b is beating c in pair wise election b is above c here, but c is above b here So b is also beating C in pair wise election But the interesting thing is c is also beating a in pair wise elections So there does not exist any Any specific candidate who is beating every other candidate in pair wise election So therefore a condor say winner might not exist But we can define something like if there exists some preference profile where condor say winner exists So, of course, this is one counter example, but you can also create certain examples where the condor say winner actually exists If that condor say winner exists then the voting rule Those kind of voting rules that returns that condor say winner as the outcome Those kind of voting rules are called condor say consistent In the name itself is saying a lot. So if there exists a condor say winner Then the voting rule will give that condor say winner as the outcome And it is not very difficult to see that co-planned is going to be condor say consistent Because of the very reason that if there exists a condor say winner That means it is winning all possible pair wise elections then that particular Winner or the candidate will have a score of m minus 1 it is beating everybody else in pair wise elections Now if you look at any other candidate that candidate will have Less than m minus 1 score because it is at least being In the pair wise election against that previous candidate who is the condor say winner It is losing against that so it can win at most m minus 1 cases, but it will lose Definitely to that candidate and therefore its score cannot beat the score of that condor say winner So therefore co-planned will give that condor say winner the highest score and that will be the winner So that's Very standard. I mean co-planned is essentially condor say consistent by its design itself But some of the voting rules that we are quite familiar with we use it quite often That is not condor say consistent and this is sort of a A fallacy in itself the plurality rule is so much used in all possible cases. It does not satisfy This very idea of condor say consistency So condor say consistency is something like very natural thing, right? So you are looking at a candidate which is beating everybody else in pair wise elections It will be very unjustified for a voting rule not to select that candidate as a winner But you you can see that in reality it does not happen. So here is an example So suppose there are again Three sets three different types of voters By type I mean that their preference profile remains the same the preference ordering Among these three candidates remain the same. So 30 percent of these voters prefer a over b over c the other 30 percent Things that b is above a a is above c And there is 40 percent which thinks c is above a a is above b right Now if you look at the pair wise election situation, you see that may a is above b In 40 plus 30 that is 70 percent of the case. So it a beats b in pair wise election by a score of 70 to 30 Similarly a also beats c here. You can see that a is above c For this 30 percent and 30 percent. So it also beats c with a margin, I mean With a margin of 60 percent to 40 percent Now clearly a is the condor say winner because it beats all other candidates in pair wise election Now if you look at the plurality rule, so plurality rule just looks at the topmost Vote see every agent every voter is asked to cast its ballot For the for the most favorite candidate and in that case a will get only 30 percent of the vote b will get 30 percent of the vote c will get 40 percent of the vote And therefore c is going to be the winner in plurality, which is not The condor say winner condor say winner was a So actually you can Prove it even more generally you can create counter examples for any kind of scoring rule So no scoring rule based mechanism that we have discussed In in in the beginning all this plurality rate over the harmony k approval None of them are actually condor say consistent. This is this is quite amazing Okay, so let us now go back to the social choice functions setup. We have this function f which is mapping the individual preference orders to this set to the set of all alternatives or all candidates Now we are going to define a few Things few notions which will be useful for the For developing them the conditions and the results involving social choice functions So the first thing is Pareto domination So we can call an alternative a to be Pareto dominated by b if Very naturally that alternative is strictly dominated by b by all the agents So b is strictly preferred to a by all the agents Now we talk about Pareto efficiency for a social choice function So we call a social choice function to be Pareto efficient If there exists is if for every preference profile where you have a dominated alternative Then that social choice function will never output that dominated Alternative as an outcome So in every preference profile, let's say a is Pareto dominated then the f of b can never be equal to a So which is quite natural Why should the society choose an alternative which is which has a better alternative than that one by all the agents So Pareto of efficiency is is the first property that we will be looking at And unanimity by the name itself Says what it is So we are going to call as social choice function to be unanimous If you have a preference profile where the top alternative of all the agents are the same So they agree on the top most alternative then the outcome should also be the top most alternative Right, so this is this is what it means that if the the top most so p P i k is the kth preferred alternative of this agent type so p One one means the it is the top most alternative the the most preferred alternative of player one Okay, so that is unanimity now. We can actually start drawing A relationship between this Pareto efficiency and unanimity already So we observe and I claim that Pareto efficiency will imply unanimity So what does that mean that if we have a Pareto efficient social choice function that should also be unanimous Why is that so we'll have to show so suppose we start with a social choice function, which is Pareto efficient We'll have to show that that is unanimous as well Now for unanimity the if condition itself is that This is going to be The top most alternative is going to be the same for all the for all the voters or all the agents So let's say that is the that is given to you that the top most alternative is the same And now my social choice function is also Pareto efficient. That means it will never take It will never output an alternative which is Pareto dominated Now because a is the top most Then a actually Pareto dominates all other alternatives So you cannot really pick any of the other alternatives apart from a because that is Pareto dominated by a So if you just rule out all the possibilities the final thing that you have Left with your left with is a and therefore f of p has to be equal to a because of the property of Pareto efficiency So which is essentially nothing but unanimity. So if we have a Pareto efficient social choice function Then we we can show that that is unanimous And we have also used this strict containment That is Pareto efficient is a strict subset the social choice functions Which are very efficient is a strict subset of the unanimous social choice functions Why is that? So you can consider I mean breaking the condition of unanimity is very simple You can of course whenever you have the top alternatives to be the same the social choice function should be equal to a But imagine a situation where all the top most alternatives are a but except for one agent, let's say p And for that a they live somewhere below and maybe there is c And c is possibly a Pareto dominated by a so For this kind of a preference profile, you cannot really apply I mean This it could be unanimous because it It does not satisfy this condition. So the social choice function can actually output something And let us assume for this particular social this particular preference profile f is actually outputting c which is a Pareto dominated thing So, uh, this is this f is not violating the condition of unanimity whenever all the top most positions are same Uh, then f of f of that Preference profile should be equal to that top most alternative But when it is not even if you puncture one, uh, one, uh, preference order of one agent you pick something which is Pareto dominated and that is That is not Pareto efficient that social choice function will not be Pareto efficient even though it is unanimous So that is one example why this is this Containment is essentially strict. Okay. The third definition that we are going to discuss is that of onto-ness So we are going to call a social choice function to be onto and this onto is just the The natural normal definition of function onto-ness That for all alternative, uh, a in a So lowercase a in the set of alternatives There must exist some preference Profile preference profile p of a let's say In that script p whole to the n such that if you apply f on that you get get back that alternative a as your output And this is also very similar. I mean the This implication of unanimity implying onto-ness is pretty straightforward. All that you can you Do here is pick the the a on top And because of unanimity, you will you will always output the f will always output for this kind of a preference Profile the outcome of a So no matter whichever alternative you pick you if you pick b You can replace all the top most positions with b and apply unanimity and you will get b as your outcome. So therefore it is I mean that f will always be onto so therefore unanimity is also going to be onto Now you can uh, I can leave this as an exercise to show that this containment is also Strict so you can have some social choice function, which is onto but not unanimous So let's try doing that Okay, so let us move on to the next definition and this is the This is the place where we are actually talking about strategic behavior. So manipulability We haven't thought about manipulability when we talked about the arrows impossibility result In particular even before that when we were discussing the revelation principle We had actually spoken about the manipulability. So let us get back to that point In this domain of social choice functions So we are going to call a social choice function f to be manipulable If there exists a player and a preference profile b such that For player for that player i if that Player is reporting misreporting its preference to be p i prime And other players are actually choosing their corresponding p minus i Then it is better for for that agent. So You know that this outcome this f of p i prime p minus i is one outcome Let's say this is a and this outcome is b. So then By misreporting it is getting an alternative which is more preferred Than the outcome if it was not misreporting So we have seen this This kind of a definition in the context of cardinal preferences and this is ordinal preferences About dominant strategy incentive compatibility. So you can just Invert or negate this statement of non manipulability in that case What you will get is that for all agents i and for all profiles p The the inequality so this will essentially flip that is you don't get so f of p i comma p minus i should be at least as good as f of p i prime p minus i that we have already defined as Truthfulness or a dominant strategy incentive compatibility here. We are defining it in the reverse way that is manipulability So when we are talking about manipulability, let's look at some examples how Some of this known voting rules can be manipulated. So again the The usual suspect is plurality. So let's say that we are having this plurality where we are just asking Each of these agents to report their most valuable most preferred candidate and we are counting how many of them are there and Breaking the tie in favor of a over b over c So let's say there are four voters who are having this kind of a vote Four voters who has this kind of a vote and there is exactly one voter who has this kind of a vote So what is what is happening is that the it is being tied between a and b and This particular individual this last individual is sort of a tie breaker Now it it sees that if it reports his vote truthfully Then according to this tie breaking rule a is going to be the winner, which is its worst preferred alternative Instead if it changes its preference to b votes for b Then what it gets is at least better than that it knows that c has no chance of winning in this priority vote and It chooses to vote for its second best candidate and therefore it is manipulable You can see this by this definition You could already find some p i prime for this particular agent such that it gives them a strict better payoff And this is something that you you you have possibly seen Or at least think about seeing this kind of a situation in real elections The second case is about co-planned now plurality. We know because this is a scoring rule based mechanism This is not kandose consistent But co-planned is kandose consistent again the the tie breaking rule We know that it is in favor of a over b over c And what we are going to see in co-planned mechanism co-planned voting rule is that we are looking at pair wise Elections and we are giving scores for that now a bits b Impair wise election, but loses to c. So therefore a has a score of one Similarly, if you look at b and c you can see that their score is also one. So if it is truthful, so if All agents are revealing their Preferences truthfully then A is going to be the winner because that's the tie breaking But now you can see that this particular individual and here there are exactly one Voter in each of these classes So E is the least preferred alternative for this agent here And if it reports, let's say c above b. So then at least it it gets c to be the winner Because in that case c will Bit in a pair wise election even b So c will be the the co-planned winner and therefore c is at least C is a strict preferred alternative than a which was going to be the winner otherwise So we can see that again according to the same definition of manipulability. We found bi prime for this particular agent which Which essentially is beneficial for it Then it's true true preference or Okay, so we are going to call that a social choice function to be strategy proof This is the term strategy proof. We are coining for the first time Yeah, we will be using this interchangeably with truthful Incentive compatible and so on so strategy proof If it is not manipulable by any agent at any profile So essentially we'll have to negate this statement the the manipulability statement and we can Write an equivalent statement Accordingly and as we have discussed for all i in in n and for every profile p The opposite of this This implication should hold So what is the implication of strategy proofness? Let us define a few things in order to In order to understand this So we define what is known as a dominated set of a specific alternative a at a Preference of bi So as the name suggests these are the set of all those alternatives Which are below a in this preference profile in this preference bi So let's say in this example we have this alternative d and this This dominated set of d under this bi is nothing but a and c which are below that particular alternative In this preference order. So that is what it means. This is the formal definition. And this is the this is an example The second the other definition that we are going to use Here is that of monotonicity. So what is monotonicity? We are going to call a social choice function to be monotone If for any two preference profiles, let's say p and p prime with the first in the first preference profile the outcome Or the the social choice function outcome is going to be equal to a And this dominant dominated sets are essentially Increasing in its size. So if we go from pi to pi prime, we see that the dominated set All the candidates all the alternatives that were below a in pi continues to be Below below a in pi prime. There could be some new alternatives which are now going below that So in some sense a is relative position is actually weakly increasing in in pi prime than in pi Then this monotonicity is saying that f of pi prime should also be equal to a and they should hold so this Dominated set condition should hold for all agents in i in n So every agent is now having more alternatives which are Which are worse than e so if we already have Outputed the the social choice function has outputted the alternative a Then why it should continue to output it Even in this new preference profile that is what monotonicity means So here is one example. So let's say this is our p So so for all the different alternatives for for all the different Agents these are the positions of a and This is in p and in p prime what is happening that they are actually Climbing a little above. So maybe a is climbing above So that means some of the other alternatives which were here Let's say b or c They are now coming down and sitting below All the alternatives which are below it are always and they Still remain below a And similarly here For all the agents but now new Possibly some new alternatives are also coming below that So a is relative position in this is weekly increasing So if so the monotonicity is essentially saying this if f of p was a so in this preference profile If your outcome was a then here also you should have a as your outcome if you apply f to it It should output a and that is monotonicity Now let us conclude with one result and we'll prove it next time If a social choice function is strategy proof Then it implies and is implied by a monotonicity if and only if condition