Algorithmic contract theory and contractible contracts with Ron Lavi | a16z crypto research talks

Transcript

good afternoon everyone uh very happy to welcome Ron Levy uh from University of bath Ron has been involved in algorithmic Game Theory and specifically algorithmic mechanism design almost since the beginning right over 20 years uh worked with Nom Nissan I guess for your PhD yep um and uh right postto at Stanford is that that count that's right that's right in EC lately there's been one thing that's really blown up over the last few years is contract theory not so much contracts directly in the sense of smart contracts in the blockchain world but in the sense of um kind of microeconomics um and Ron's been uh doing some really nice work there so he's going to tell us about contract theory both in the traditional sense and also maybe connections to blockchain contracts so run the talk will be mostly

based on a joint work with Elisha shamash from Tech since the early 2000s uh we have seen uh many tight connections between economic theory and uh the new computational computational reality that has evolved so significantly uh in the past two decades most notably we have seen these connections uh um through auctions uh and auction Theory um so it was a two-way connection so it was not one thing influencing another but actually it it was like um a relationship one object influences the other um so the new computational reality back then the

internet electronic Commerce all these uh Hightech companies created new there was a new computational reality and this uh created New Economic models results mathematical techniques um and there was also the other way around um economic theory established economic theory established models um um sparked new algorithmic models results mathematical techniques and what I'd like to to do today is to uh argue and hopefully also demonstrate um that a similar possible um connection two-way connection could be

developed uh between the classic economic theory of contracts or maybe also we can algorithmic contracts and um this new object of smart contracts which to me uh represents uh one of uh the major Promises of of of the new uh the current computational reality which is the cryptocurrency systems so on top of all the uh contributions to finance um really um there is also the potential promise of smart contracts as as as an object that could influence other reality computational realities and and physical

realities so so if you look at the situation today it very much resembles what was back then in terms of comparing auction Theory and auction reality so if you look at economic contract theory today um so it's mostly irrelevant to uh the new cryptocur cryptocurrency reality and so if you try to see if there is any connection then most probably you will just say that it's completely separate to different um disciplines um or or objects um um but however um um I I would like to try to to to to claim that this is not the case

and in fact uh we could make this Theory we could evolve this Theory um into into a new one relevant one um that will have this two-way connection and and will enable us to to demonstrate how smart contracts can influence um many physical realities outside of the finance world so in order to do that I'm going to to to abstract way and and describe a general model of of of contract theory which will initially look irrelevant and then I will describe uh this this this issue of contractable contracts which very much

resembles smart contracts and hopefully uh this will in um motivate or Inspire or whatever um um to to but of course this is just a very beginning and so uh this is just a a hypothesis an argument so this is a very abstract model a model of multi agents and multi principles um of course a contract is is is this object that is being signed between uh two parties one party is called the agent and the other party is called the principal um and we could even think of contracts that are signed between multiple agents and multiple principles

and so In This Very abstract framework um we have agents and uh the agents um will need to choose a costly action out of a set of possible actions so each each agent has a set of actions to its disposal and um they need to choose one of of these actions and actions could be costly and then there are the other side which are the principles and each principle obtains a a gain from the actions of the agents so so the the gain of of a principle is just a function of the combination of actions of all the

agents um and principles may have contradicting gain functions so some principles want the agents to go left While others want the agents to go right this is the model that for example was studied in by pra in rasini in 2003 we could think of many uh realistic applications of this model um for example um um online labor markets where Freelancers complete tasks posted by platform's customers so Freelancers are the agents they need to take some action um and the customers are the ones that are the principles online advertising where a social media influencer for example promotes several Brands a marketing

agency that uh bids for ad space on behalf of multiple advertisers platforms like Airbnb or or booking.com represent multiple property owners so all of these are cases where we have uh one side which is the principle try to motivate the other side which is an agent to act on their behalf and the Agents represent multiple principles and the principles interact with multiple agents so we could also look at it as a in an example consider the following example so in this example um the rows and the columns represent the possible actions of two agents so we

have two agents the row agent agent one has two possible actions uh action t or for top or B or for bottom and the colume agent uh also has two possible actions either left or right and in this Matrix I describe the utilities that the principles H get from the combination of actions of the agents so note that this is not a usual game as we uh um sometimes see in Game Theory 101 because the numbers inside these cells are not the gains of the agents

they are they are the gains of other entities these are the principles so if agent one takes T and agent two takes takes L then there are two agents sorry there are two principles one of the principles principle one will gain five and the other principle will gain zero and so on so we have four possibilities because that's that's the example you should also notice that principles have conflicting preferences regarding uh the agent actions so principle one prefers TL principle two prefers either BL or or or t TR um and um yeah and and so on so we have a game here so we we can define

a game here where in the first phase um every principle announces a a contracts which are payments uh from the principal to the agent based on the action that the agent chose to to take right so that's simply a function from the action space of the agents to the real numbers if you as the the principal one tells agent one so this is like the T subm subn principal M tells agent n if you are going to take a certain action I'm going to pay you a certain

amount so that's the first step of the game the principles announce contracts then in the second step of the game agents observe contracts and decide on their actions and then we have a utilities being realized um um the principle of U utility is the gain from the combined set of toule of actions minus the sum of payments and the agent utility is the net gain so it's the sum of payments from all of the principles minus the cost of the action so this is the agents utility which is not present in the in in the picture but yeah and it depends on the

contracts so for example if you go back to to to to this example um then we could see the the following possible contracts uh principle one could offer zero for t for agent one one for B zero for L for agent 2 one for R so with this contract principal one incentivizes the two agents to play BR and if principal two just does the same then with these contracts the agents will indeed so and I'm assuming for Simplicity that the costs are zero of of all the actions just for the Simplicity of this example um and so with these two contracts uh

the agents will indeed uh play BR and the gain of every principle will be three and the payment of every principle will be one to agent one one to agent two and so overall the the principal utility will be 3 - 1 - one which is equal I think one so given the contracts yes the agents optimization problems are completely decoupled right like as a yeah exactly given contct you're an agent I don't care what you do yeah I don't care yeah yeah agent one doesn't care about the decision of agent two okay so you can think of that is happening simultaneously independently given the contracts given the contracts you have to discuss the timing of the

choices of the principal contract yeah I do have and and and in this specific paper I assume that this is simultaneous so we have two steps in the first step the principles announced simultaneously in the Second Step the agents announce act simultaneously it does does it actually then just become a in your example two-player game like just a game noral sense so there are four players here two principles and two agents right but can't I just sort of wrap the agents up into the so if I think about the strategy space now being contracts yeah so so just want I have a game with two players the principles strategy SP is now the contracts yeah given a choice of contracts I just fast forward to what the agents will do I now know my payoff

but it's it's not that simple because so for example if you think about the the action space of the principles each principle needs to submit a contract which is which contains four numbers and so it's it's it's it's not really just a choice of no but it's a game right it is a game between the principles and then it's it's and then the the agents act upon the incentives that the principles create so can I think of it actually yeah you can you you can and actually that's of course that's a very good point and in fact if I go back to this um credit so the the title of the paper by pratton Rini is games played through agents so that's

exactly what you were aiming at so it's games that we don't play directly we play through agents so I agree with you but but I think it's also a little bit misleading because really it's the principles is motivating the agents and it becomes complicated because of of that of that fact this is exactly the point of this picture to exactly demonstrate that it's a game played between the principles through the agents you might notice at this point that um these are not very good contracts for the for the principles and and specifically each principle can increase uh her own utility by unilaterally submitting

another contract so for example a principal 2 May observe that if principle one is already motivating the agents to play BR then there is no reason for principle 2 to also spend money and motivate the agents to do the same they could deviate to um zero to to just offer zero on all possible actions um and so so so there are a lot of possible deviations uh one deviation is is for either principle is to to deviate to a contract that offers no payments three writing on the other principle principle one can incentivize the a the agents to

I mean wants to incentivize the agents to play TL um and principle two would like to incentivize just incentivize just one of the agents to move from for example agent agent two to move from R to L um and so on so there could be many possible deviations and in fact Pratt and Vini gave a characterization that can be used to show that in this example there does there does not exist an equilibrium so when whatever contracts you think of um there there will always be a profitable deviation for the other principle and there does not exist a pure equilibrium in this simple example so what were they characterizing like

what they they they gave a characterization in terms of uh a linear program uh um and it's not a very crisp characterization that I could actually describe so it's it's it's it's a very technical condition um and and but if it if it holds then what I mean they gave sufficient and necessary conditions for for for for for the existence of efficient equilibrium of efficient equilibrium yes of efficient equilibri got it so so when you say imply there does not exist a p you mean there does not necessarily exist a pure equilibrium no in this example there does got it so sometimes you'll have one

sometimes you won't in this case you won't yeah in this specific case two actions 5 z04 and they were characterizing existence of a pure and efficient equilibrium or just efficient equilibrium so so they they give us a technical tool with which we could check whether there exists a a pure equilibrium in this in this example and then one of the clarifing questions so like this is still like if you only had one agent this would still be pretty non trivial is that right multiple principles One agent like CU there still be this nontrivial game theory amongst the princi no so if you had just one agent then then there is even a previous I mean like one principle multiple agent

seems not that interesting right right but like one agent and multiple principles so I think you could you could look at it as if it is like an auction the agent is is so I mean each principal wants the agent to take the action that maximizes the gain because the there's just one agent so each principal has yeah yeah so I think I think you could there some okay there's so there's still some Game Theory left but some it's definitely simpler than this okay yeah if there yeah so I'm pretty sure that if there is just one agent then it's going to be I mean equilibrium existence is is is guaranteed and I

think you could always come up with equilibrium which is efficient and so yeah okay I think there's a paper from the 80s that that describes that it's called common agency One agent and multiple principles right but the whole point of what I want to talk to today is the fact that with multiple agents and multiple principles there does not exist a p pure equilibrium and and and this is my starting point so um our idea in this work is to allow more sophisticated contracts and in particular we would like to Define contracts that depend on other contracts and this these are termed contractable contracts I could

only find two previous papers in uh the literature that actually Define or use such an idea uh in the economic literature but of course in reality we have Smart contracts which exactly enable us to do this but there are other examples we don't have to only think about smart contracts um for example um if you look at Hilton's web page uh Hilton has a price match guarantee uh we always do our best to offer you the lowest price but if you find a lower qualified price on another site um we'll be happy to match it and give you a 25% of your state so if you

buy a a hotel now with Hilton um you're not exactly sure how much you'll pay you know what's the the best price the worst price the highest price that you'll pay but it's not clear if you will find a lower price then then you get a lower price so this is another example of a contract that depends on other contracts U if you could find a better price on booking.com then we'll match that price so about things being well defined now right sorry you have to worry about things being well defined yes of course you have to worry about things being well defined because it could be a recursive definition yes yes yes absolutely but but this holds also

for smart contracts you know another example this is from the UK there are two supermarkets wellknown supermarkets in the UK Aldi and sabes Aldi is a lower range prices and saintsbury is is a higher range prices and sabes give a price match guarantee to Aldi Al the price much if you're going to buy a a product at sainsbury you don't you know the the an upper bound on how much you'll pay but if Aldi offers a lower price they'll match it so this is so this is the idea that I want to um um advocate in in this presentation uh to explain how a contractable contracts and the specifically a very specific and

easy form of a contractable contract can uh resolve the inefficiency of the previous game where there does not exist an equilibrium because of the structure of the game H and turn it into a an existence result um and and and this is this is exactly the highlevel contributions that we saw in the early 2000s regarding auction theory that um um demonstrated how a a theory that on the face of it looked completely irrelevant to the new computational reality somehow turned to be very relevant and so back then they used all these abstract models that even

though they were abstruct turned out to inspire actual constructions in the physical world actual things that happened um because of all these theories that interacted and so that's that's I hope um would be my my conclusion so I want to explain how to H um or to give intuition why I I I have hope that contractable contracts will be able to uh resolve the issue that we saw previously uh in that uh in that simple game simple example that we that we analyzed and so to do that I'd like to

go back to the uh world of auctions and to consider um this this classic setting um of simultaneous first price auctions so this is like I think maybe 10 years ago and U team had many contributions in that setting uh we have sellers uh each seller is selling an item and we have H biders um and each bidder is uh placing a bid on every item or not on all of them but on on subset of the items possibly and we have we have many biders so so the sellers receive many bids and then they need to allocate the item to

one of the biders and in the first price auction they just give it to the to the bidder with the highest bid um and I want to argue that this is very actually very similar to a to a to a contract scenario it's a special case of a contract scenario because if you notice the bids are actually payment offers um which is exactly what principles are doing in the contract work so so the the the entities that are offering payments are the principles so we could call the payment offer contract and then the entities that offer the uh the contracts are the principles so we could we could replace this with principles and um the entities that

receive the contracts the payment offers and act upon that are the agents and the the biders offer multiple bids so they value a uh subsets of the items or subsets of the actions of of the agents um and so you could actually be more formal and show that this actually translates to be a special case of the previous model that I was just telling you about uh where the agents are the sellers the actions are which uh Which principle or which buyer or bider to give the item to um and the biders are the principes and the BS are the

contracts so so it's special and the agent cost functions are zero right yeah the cost are the cost are zero is it otherwise otherwise special because um the biders don't care about actions of Agents other that whether I got the item or not so if I didn't get the item the bidder doesn't care if the item went you're saying the principles use contracts where the payment is a function only of that agent's action and not of the full action profile yes yes exactly yeah yeah yeah so it's it is a a very special pie on on one hand but on the other hand if you move from first price to Second price auctions then this

becomes a completely different story because now I cannot claim that these bids are contracts because when a bidder submits a bid to a seller we have no idea what will be the actual price the price is no longer encoded in the bid the price is encoded in the bids of the of the other biders to the same seller and so we have here a case where the contract of the bidder to uh the agent also specifies some dependency to what happened elsewhere and so um this is not a

special case of so the the second price auction is not a special case of the previous model for example the pr model and so the whole purpose of of the whole technical purpose of of this work is to define a different type of contract that will make it a special case that that will make both the first price and second price a special case of the principal agent uh setting so that's the motivation I think and and let me just give you a very high level highlight of our results so we Define a new type of contra that depend on other contracts these contracts significantly expand the set of efficient pure equilibrium out outcomes with classic and in particular it contains cases where classic

contracts do not admit equilibrium outcomes and uh the new contracts will admit equilibrium outcomes and we will also study the price of Anarchy of the game because we increase the equilibrium outcomes and so we have to be careful that we don't introduce very bad outcomes on top of enlarging the set of good outcomes and so this can be captured by price of Anarchy analysis um and and we argue that they are still relatively simple and all these results hold even if each principal is allowed to choose which contract to submit either a classic contract or the new contract and so um even if the principales have choice they prefer to

go with the with the new contracts of course they could have a much larger choice they could have many types of contracts and then the problem becomes even more difficult but that's exactly my point the point is that algorith a classic contract theory is so limited in its real its assumptions the way that it is constructed relative to the new computational reality that if we start studying contract theory with the actual tools that we in light of the actual tools that we currently have then um we will be able to demonstrate how these new types of contracts could help not only financial entities but also

booking.com Airbnb sabes ilton all these physical entities so recall the game uh we have a principles uh that announce initially it was classic contracts but I changed the notation from T2 B and so the the the B which like stands for bids because of my motivation of first price versus second price auctions um so these are you could think of it as the maximal willingness to pay uh that's or or in the case of a classic contract this is actually how much they will pay and then after observing the contract the agents simultaneously choose the actions and then we have the utilities realized again as before gains which the gain is a function of the combination of actions

minus uh payments and the agent utility is just uh the uh the sum of payments minus the cost by the way I just for simplicit I will because I'm going to do some some some very light writing of equations I don't want to drag the costs with me but in the paper we actually do drag the costs uh all along uh right and I'm going to change that to what we call vcg contracts uh where we don't have these B we have more General payments that depend on the action but also depend on the other contracts in a very specific way that I will mentioned soon the the utilities are not the the bids but the payments

and minus with the principles plus with the agents sort of I mean motivated by your second price auction example I could imagine a strong motivation for restricting the dependence on T minus M to be such that your payment to an agent little end depends only on others contracts payments to agent little land and not on the others so you depend oh so you're saying this is in general this is capital T but you're saying it will capital B well no I'm saying like yeah I yeah this is what we will do actually like in the second price auction one there's dependence the dependence is pretty limited it's kind of like local to each agent right right going back to is that sufficient for your positive results yeah it is sufficient I think I think so if I understand correctly what you're saying you're going back to your

concern about the well well defined well I that's a sep thing this is more like a kind of simple versus optimal argument right I mean you could imagine that or like if I'm an agent and I'm getting paid less than I thought thought right and then and you say oh well that's cuz these other guys sub these contracts it's nice if you only point to the other contracts and how they relate to this agent whereas this where allows kind of full depend I see you see what I'm saying so like you're allowing the transfer to agent 6 to depend on other principles contracts for agent nine which is a little weird right potentially so yeah it it would only depend on the contracts that the other principles submit to that agent so you only need that yeah I only need that

that's great yeah yeah yeah I only need that great yeah but in general yeah let's if if yeah okay I don't want to confuse the others at this point too much it's just about the payments now will depend on the action but also on the other contracts because I I mentioned the word wcg let me give you a very high level description of vcg this is more than enough it's actually strictly more than enough but uh I still want to to do it for intuition purposes in mechanism design in General our goal is so there is a mechanism designer which is like an impartial entity that wants to act for the favor of the society and the goal of that entity is to choose one alternative out of a set

of possible Alternatives so that mechanism designer has to its disposal a set of possible Alternatives and um they want the mechanism designer wants to choose one of these Alternatives based on how the uh players value these Alternatives so we have players we have many players and each one of them has a value for each alternative so player one has values Vector of values for Alternatives player two has a vector of values for Alternatives and so on the way that that a mechanism in general works is that the players report their values all these numbers to the mechanism designer and then the mechanism designer chooses an

alternative based on their reports and the mechanism designer can charge payments to incentivize players to reveal their true values because this is a game and once again the players need not necessarily reveal their true values and so the output of the mechanism designer is the chosen alternative and the player payments so each player pays like a price right and so this is like the general framework of mechanism design and with vcg we just have a specific formula for the payments um and we know that this formula causes the payments to declare the true or incentivizes the players to declare their true uh values if what the mechanism designer does is to choose the

alternative that maximizes the sum of values of the players so if the mechanism designer sum sums up the values of alternative a the values of alternative B and so on and chooses the alternative that maximizes the sum of values then the vcg payment formula incentivizes the agents to declare the True Values so this is vcg this is mechanism designed you you don't you only know you only don't know the formula um but you don't really need to know it but in any case I'll say it in the in the next uh in in a few minutes um and and in intuitively only in terms of motivation uh we are inspired by this framework viewing the mechanism designer

as the agent the alternative that the agent has is the actions that the agent has and the players are the principles and the principles have values for actions and so the principles submit values for actions to the agent the agents choose an chooses an action and then gives an output I'm sorry that that's the output the action and that's that but of course notice that this is a very very uh shaky motivation and there is really no Formal Connection to vcg and this motivation is is probably even flawed because of several reasons for the first reason is

that the agent is not a mechanism designer it's not an impartial entity the agent is a strategic player that wants to maximize their own payoff um rather than to maximize the the principles sum of values so that's the first difference the second difference is that the principles is it's not only just one agent there are multiple agents and the principles pay all of them and so it's not just a single mechanism designer there are a lot of Agents so that's the second reason and the third reason is that um when when I say True Values what what do I even mean there are no True Values because in the mechanism design framework we have we have a values for act for Alternatives

but if I view Alternatives as actions and we have multiple agents then the the the what's the value of the action T I don't know it depends on what the the other agent will do and so there are no that truthfulness is meaningless there are no values for actions so it's really a very flawed uh comparison um but um intuitively it helped us uh to formalize some solution which I don't even know if that's the optimal solution uh but at least that's that's like one step uh forward um right so that's that's the intuition for the vcg contracts and now given this intuition I'd like to uh Define exactly the vcg contracts um so

so a classic contract again is this function that gives a number to every action uh and you could think of it as the principal's maximal willingness to pay for that action and then we have a specific uh function which which depends on the classic contracts and also depends on the actions of the agents this function determines the payment of principal M to agent n and vcg contract will be a specific form for this function T and we had the intuition from before the agents are viewed as the mechanism designers and the principles are the players and the B's are like their

valuations whatever this will be um and um the principles pay the the agent the vcg payments right so because of this third bullet we could attempt and write what is the vcg payment so we could Define the actual function T to be the vcg payment like taking the vcg formula and trying to see if that will work here um and so that's so so the vcg formula has a very catchy uh um textual explanation verbal explanation uh each each principle pays to the agent the damage that the principle causes the other principles uh

and that damage is is more formally defined to be the difference between two terms the first term is the aggregate value of all the other principles from a given action that the that the agent does that's the first term and the second term is the value uh of all the other principles from the action that the agent actually takes sorry so so the first thing is the maximal possible value for the others and the second term is the actual value of the others so the difference between the maximal possible and the actual to the others that's that's the damage uh that this the

existence of this principle H caus caused the others so that's the difference so that's that's the formula H and as you could expect actually this this doesn't work there's no reason for it to work it's not a vcg there's no connection to vcg so this is really funny but it doesn't work but something luckily something very similar did work so instead of this vcg formula uh the thing that did work is is is a very related formula where the first term remained the same the second term uh was amended so that uh we add so it's no longer on sum over all the other principles I we also added the the the

the bids of of of this principle and we again maximized over all possible actions and then we added um the uh the the um to the payment we added the uh the bid of that principle to that agent for the chosen action so this is like a formula that is related to the vcg payment it's not the vcg payment if a equals a n exactly so so so I have I have two uh two uh types of intuition to offer you the first one is a short one so I will that's exactly what you just said let me repeat it um and so the second formula becomes equal to the first

Formula in one specific case in the case where the maximal among the maximum among all the A's here is exactly equal to the action that the agent takes if the action that the agent takes maximizes the possible sum of actions sum of bids for that action then this maximum a becomes a n and this cancels out because there's a minus sign here and so this cancels out and everything that we did goes back to the vcg payment right in other cases it does not so it's not always the VG payment it is only the vcg payment in a specific case but this this will turn out to be the the the the

the case that happens and to explain that um I will skip to the second slide so I'm a little bit I have I don't I I can't do it more slower than I already try to do because we have this time limitation but I'm happy to answer questions later on um and this is done completely separately agent by agent right this whole this is done per agent right but this is also like the general definition so we Define it as as BN here so it's only the bids of that that that this agent n got yeah yeah so this is like yeah this connects all your previous questions as well yeah right so that's the payment formula if I

look forward at what the agent will will do given a set of contracts initially it could seem unclear what action will the agent take remember that in classic contracts it's very simple the agent will simply look at the action that maximizes the sum of bids from the other from from all the principles because what the agent will get is simply the sum of bids uh from all the agents from all the principles and so uh in classic contracts the agents chooses an action that maximizes the sum of bids so this is exactly in classic contracts sorry the agent exactly chooses the action that maximizes this what will be the action um that the agent will choose in

in vcj contract so the agent will simply choose the action that maximizes the sum of payments it's no longer the bids it's the payments and we if but if we write the formula for the payments so I just did copy paste we could see that the the first two terms are constant because this is maximum over all actions so it doesn't depend on the specific action that the agent takes the second term again doesn't depend on the action that the agent takes so it's a constant so it only depends on the bid even though the actual payment will not be just the bid it will be the bid plus minus something

and so this is like going to be a constant plus the sum of bids that's so it's going to be exactly like in the classic contract both in the classic contract and in the vcg contract the agents are going to look at the bids and are going to is the action that maximizes for them the Su of bids if we go back to the formula the agents are going to choose the action that maximizes the sum of bids so a n will be this a that maximizes this and therefore as as team said this cancels out and we go back to the vcg payment right so so this is the these are the vcg contracts um I hope I managed to give you a little bit of

intuition uh how they work and so the vcg contracts don't change agents incentives they will still look only at the bids and will choose the action that maximizes sum of bids uh it does change the incentives of the principles and then they will generally pay less because this the maximum over here is larger than the maximum over here um and so if you interpret this as the maximal willingness to pay you will actually they will actually in general pay less and let me again tell you about our results um without any technical details so the first result says that every pure equilibrium with classic contracts will also be an equilibrium with vcg

contracts so if we had if you have a couple of contracts of bids that are equilibrium for classic contracts it will be also equilibrium with vcg contracts but there exists cases in which no equilibrium exists for classic contracts but does exist there is exist an equilibrium for with vcg contracts and these are not just a a rarity the the the measure of such uh cases which by cases I just mean gain functions uh in the space of all gain functions the measure is positive but we don't really know how much is the measure we don't know if

it's 50% or 10% or whatever but you know if it's 100% no it's not 100% that's for sure yeah unfortunately yeah just so so I understand the first statement yes so you're saying like if you take a pure equilibrium with classic contracts you keep the contracts exactly the same but now you let people deviate if they want to any of your very rich basic contracts not necessarily vcg contracts no no no you keep the space of contracts to be vcg they only deviate to by submitting different bids so the space of con so in this result the space of contracts is just vcg contracts so the the the actions of the principles is just to submit different bids but the translation of bids to payments depends on on on on

this framework okay but there aren't any even but if you allow them to also okay now I'm confused so for as a vcg contract you can mean two things you can mean I allow tmn functions of that general form or you could mean that specific payment function of this is enforced in in in what Analyze This is enforced so the game only gives them freedom in submitting the classic so so a vcg contract has the classic contract and the payment function and and this is a payment for the in the analysis we we assume that this will be the payment based on the contracts submitted the bids submitted and that's a good question I I

I completely agree with you that this should be gen generalized we did one step forward and we increased this space to also include the payment function which is just the identity so the function will be just the Classic contract and so they won't deviate to that as well but of course and if you have Smart contracts in mind then they they are of course unlimited and so it's not clear probably if not correct that that will be an equilibrium anymore VG contract contracts and I don't know what will happen so that's that's an excellent question yeah but we did enlarge so with classic

contracts you know I mean if you if you have so this is exactly what I mean if you have Smart contracts fixed already in your head then you don't even accept this notion of a classic contract why are the principles limited to submit just smart contracts that do not condition on any external events doesn't make any sense right so we make a very small step forward to increase the space a little bit and analyze what happens in a specific way of course so that the the Improvement will be clear right so so so back to your question um so it's equilibrium with classic contracts is an equilibrium if

we change the payment function from classic contracts to vcg contracts it also it is also an equilibrium I I will have it later but maybe I should say it right now that it's also an equilibrium if there there is a first zero a Step Zero each principal can decide for themselves whether they want to use classic contracts or vcg contracts they will use they will always use vcg contracts so I mean intuitively is it like basically everything stays the same except you might get an additional rebate as the principle if you switch to princip no not everything stays the same so you with so so let me answer your question after I describe VI this this picture so in the blue picture the there are every every like virtual dot on on

this on this space on this large black space is a a tle of gain functions in the virt in the blue circle the smallest Circle these are all the TP of gains gain functions games for which there exist an efficient equilibrium just replace SP with equilibrium I that's I missed that efficient equilibrium when you have classic contra in the green circle you have uh these are all the gain functions where you have efficient equilibrium with um vcg contracts and and there could be other cases as well so guess so just inside the blue circle it was just it back to that first result again where you're

just saying every equilibrium I mean your proof actually says like take something in the blue yes keep the contracts exactly the same switch the payments your V payments and it stays in equilibrium right yeah it stays a stronger State not for everything but for for a large a larger class of no I'm just working in the blue I'm only working in the blue here right so I'm just inside the blue yeah so it's the it's your argument that the blue is inside the green the argument in the blue says yeah everything stays yeah if if I was asking is the intuition basically that just like from the principles perspective if you switch it's just like you're pay you get some rebate so you said cuz you also said you always prefer vcg to Classic a good question my guess is like the way the proof goes is you're like look if

you switch from classical to vcg you're just going to get a rebate compared to where you were before no I actually think that there will be no Reit it's going to be exactly the same I think it's just going to be exactly the same so so it is just completely strategic yeah I think that if there including for deviations so you need to show two right so ver I'm not sure have to think about later yeah I'm not sure but let me also say the other two results so we have increased the number of equilibria even inside the blue Circ there could be other equilibria so so we could increase the price of Anarchy even significantly but we show that that's not the case we couldn't exactly pinpoint the price of Anarchy and of course it was not studied in in for for classic economic contracts and so they

don't even know what was what is the worst case the only thing about the best case um so it turns out that the price of Anarchy for both both of the contracts is is is about the number of principles um either that or plus one um so it doesn't decrease in the worst case yeah and I already said the last bullet that if you have pure equilibrium with BCG contracts it's also an equilibrium in an expanded game where the principles can decide whether they want vcg contract or classic contract so let me just summarize um so we study a new type of contracts uh temp vcg contracts where payments depend on bids submitted in other contracts as well as on the

principles own maximum winning need to pay um this has been termed contractable contracts by two other papers PS and sentence and sentence that's the only two papers that I know in the literature and since this paper is is already published I I I assume that these are indeed the only two papers that have this idea of contractable contracts if compare to the reality where we know that contractable contracts are being used and then uh that's that's a huge discrepancy right so that's in terms of the motivation um um we show in terms of results we show that vcg contract expand the set of efficient equilibria in a meaningful way

um even if principles are allowed to choose between classic and VG contracts the additional equilibria do not increase the worst case price of Anarchy I also have a follow-up paper again with alisheva and with Inbal and her um student Talon which and that paper studies vcg contracts for a common common agency setting with more Hazard so in the in in the the in the new newer setting um the principles cannot observe the agent actions they only observe the outcomes and they have to infer the actions for the outcomes so that's more complicated and the results are more complicated and of course we have many future directions uh for example as already discussed other types of useful

contractable contracts better price of Anarchy wider equilibrium existence for other types not for for VG contracts um and and maybe also actually thinking about how to implement contracts from contract theory um using smart contracts there are there are issues that are not dealt with like the complexity of different types of contracts the cost of different types of contract and so on and um and that's that guess at a high level so you said like the principles they do it simultaneously right like they release yeah so do you see extens to like cuz like if you do smart contract it's not going to be

simultaneous right so like what would change in like yeah just in general like that's a great question by simultaneously I don't mean that they look on their clocks and step a step forward exactly the same time but they determine their contract without knowing what the other principles determine sometimes this is correct sometimes this is not correct I agree um but it's not necessarily incorrect in in in smart contracts because I don't know if there I don't know if there's a deadline and we can we need submit something and we submit it very close to the deadline after working on it for for for a week then it's the

Assumption of simultaneously approximates reality um sometimes I agree with you that the Assumption of simultaneously doesn't agree with reality I have some this is I'm called online algorithms of course and I I have some yeah results on myself but makes everything very much more complicated so I mean for me I think this is like a simplification that I'm willing to take right now because there are other issues that I feel are more important but I but but that's that's always a situation that is worth and attention at some point in the blockchain context we often talk about enforcing that through cryptography where where

you sort of commit to aisc version of your contract and then you reveal it only after both people have committed and there's other issues that come up but like that's one way to right thank you thanks to get your information assumption thanks okay one more quick thing uh so like usually here you said like I guess for every agent it's basically independent right like once you see the bids uh like what the principle sort of bit for other agents is independent in some sense so I was wondering like are the scenarios where that's not true like let's say the principle sort of only gets a gain when it gets 10 agents to do what it wants them to do just curious in general like if that's something Ian I could imagine situations where if multiple agents I mean where actions could interfere with

one another so for example the agents could care about the actions of the other agents because there's interference yeah yeah I mean if if you look at what happened in the early 2000s and and the the the volume of new models uh that that were inspired by actual difficulties in reality trying to align reality and Theory so I I I I think much something similar should happen also with contract theory yeah I guess more broadly sort of computer scientists found out about mechanism design like 20 years ago and then all of a sudden there was an explosion of work right and even though they're equally classic for some reason computer scientists only found out about

contract theory like a few years ago so we're like 3 years into what will be possibly 20 years exactly yeah I agree yeah no and and and even more than that what I argue is that in in contrast to what happened with with mechanism design auctions where computer scientists learned about the model but then discarded all previous results results started from the 70s and then gradually build their own way up what we see now and this is like a world of criticism what we see now in computer science is that we adopt the classic Model as like a holy grail and just study computational issues you know how is it comp how much is it complicated to calculate a best response

in the classic Model or something like that so you think computer scientists have grown too differential of Economist in the last 20 years to to to the opposite of differential to yeah to respectful to Too Close so we got to get back to like yeah to that SP okay cool I mean yeah there's contract theory it's very it's cool but we we yeah we could let's make it our own you're saying yeah we could make it our own yeah exactly cool thank you