pareto dominance This is a topic that many people are looking for. khurak.net is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, khurak.net would like to introduce to you Game Theory 101 (#52): Pareto Efficiency. Following along are instructions in the video below:
“Back to game theory 101. I m william spaniel. Today s topic is pareto efficiency. Efficiency.
Let s start off by looking at a couple of games. This is a of battle of the sexes. There are two pure strategy. Nash equilibria balle balle and fight fight.
I want you to take a look at those two equilibria and decide which of the two is more efficient balle balle or fight fight. If you need to pause. The video now. But if you have an answer in the comment section.
Below the video go ahead and write. Which one you think is more efficient balle balle or fight fight don t submit it quite yet. Though because we have one other question ahead of us. And if you re ready for it.
Let s take a look at that this is another version of battle of the sexes. There are still two pure strategy. Nash equilibria ballet ballet and fight fight. I want you to look at this game and decide which of the two equilibria you think is more efficient ballet ballet or fight fight if you need to go ahead and pause.
But if you know your answer type it out in the comments section. Maybe give a brief explanation as to why you picked one of the two options in each of these games and then go ahead and submit that comment and if you re ready. I will reveal the answer now it was a trick question these games are identical. Why are these games identical well if you remember back to last lecture.
When we looked at positive affine transformations. We found that expected utility representations of preferences are identical across an affine transformation. A positive a fine transformation. Where we take the original utilities multiply them by some number alpha or a greater than zero.
And then add any other number b to each of those utilities..
And that s exactly what i did between those two versions of battle of the sexes. So if we were looking at this original game right here. I m guessing that you probably said. Fight fight.
Was the more efficient of the two equilibria and your justification was that if you add the utilities for both players one hundred plus. One that s a hundred and one and that s substantially larger than the utilities for ballet ballet. That s one in a two that s only three so fight fight appears to be more efficient in this case. But when you get to this other game well now balle balle may appear slightly more efficient.
If you use the same logic. As before adding up those utilities gets you two point zero. One whereas fight. Fight gives you two so maybe.
It s a little bit closer under these circumstances. But by that same metric balle balle appears to be the more efficient of the two yet if we look very closely. What was going on here. It was a positive a fine transformation back to the original game right here take a look at player 1s payoff.
We have zeros 1 and 100. If we take all of those payoffs and divide them by 100 just for player. 1. That s a positive affine transformation and that leaves us with the game that you see here.
This is the second game. Where now the zeros remain zeros. The 1 turns into a point 0 1. And the 100 turns in to a 1.
So this is the same game. It represents the same preferences for player 1. So whatever we thought about efficiency for the first game. We really should be thinking about efficiency for the second game.
As well what this is revealing is a problem with something known as inter personal utility..
Comparisons because utilities represent preferences identically across positive affine transformations. We cannot compare utility quantities. The exact numbers that we re using between players that s because we could always take a single players payoff and multiply. It by some arbitrarily.
Large number or equivalently add a very very large number to it and suddenly that player s utility is going to appear to be larger even though we re still representing the same exact preferences. Nevertheless we have some sort of sense of utility in game. Theoretical models that we ve looked at before for example. Let s look at the prisoner s dilemma.
This was the very first game. We ever looked at and we saw that there was a clear inefficiency going on here. We know that the equilibrium of the game is mutual defection. So both players get a payoff of 2 and that was disconcerting because there s this alternative outcome.
Cooperate cooperate that leaves both parties better off both players could receive a payoff of three for cooperate cooperate. Whereas. They only receive a payoff of two for defect defect. We ve also seen other similar sorts of inefficiency.
This isn t a stag hunt stag hunt had two pure strategy. Nash equilibria stag stag and hare hare and the stag stag outcome seem to be more efficient in some way than the hare hare outcome again both parties were better off with the stag stag outcome than the hare hare outcome. If we tweak these payoffs just slightly. We make player two s payoff for hare hare.
3. Instead of one there s still a similar sense of inefficiency going on here with the hare hare outcome. Yes. If we compare a stag stag to hare hare player.
2 is getting the same payoff. But player 1 could be made better off by switching to stag stag. Then leaving it as hare hare and that s where this notion of pareto efficiency comes into play. What we can do is take the general idea from the last three games and the three uses of efficiency in those contexts and formalize it into something known as pareto efficiency.
So we say an outcome is pareto efficient..
If there is no other outcome that makes at least one person better off without leaving any one worse off and similarly. We say an outcome this pareto inefficient. If there is another outcome that leaves at least one person better off without making anyone worse off let s see that in action with some of the games that we ve been looking at earlier. This is our good friend.
The stag hunt. We have a single outcome in the stag hunt. That is pareto efficient that s the stag stag outcome. Under this outcome.
Both players receive a payoff of 3. That s each player s largest payoff for the game. So we can t switch to any of the other outcomes. Stag hare.
Hare. Stag or hair hair and make at least one party better off without making anyone worse off any switch is going to make both parties worse off in fact so stag stag is the only pareto efficient outcome in this game. And it just so happens to be an equilibrium. 2.
Which is not there can be games with multiple outcomes that are pareto efficient. The prisoner s dilemma is one of them and this is actually somewhat disconcerting. There s only a single outcome that is pareto inefficient in the prisoner s dilemma. And that is the unique equilibrium of the prisoner s dilemma defect defect.
This is pareto efficient again because we could switch to cooperate cooperate. And that would leave at least one party better off in fact. It will leave both parties better off. And it s not going to make anyone worse off.
But the two other outcomes cooperate defect and defect cooperate are also pareto efficient. The logic is the same for both of them so let s only look at the defect. Cooperate outcome in the bottom left why is this pareto efficient well notice that player. 1 receives a payoff of 4 for the defect cooperate outcome.
That s his best outcome..
He can t do better and in fact switching to any other outcome makes him strictly worse off you get a 3 could get a 1 he could get a 2 anyway of those or any value that he s getting other than the 4 is going to be worse than the 4 that he gets by defecting so you can t look at any other outcome and make at least one party better off without making anyone worse off player. 2. Is going to be made better off by switching that one to something larger. But because player 1 is going to be made worse off.
We say that the defect cooperate outcome is pareto efficient and like why is the cooperate defect outcome is pareto efficient so the three efficient outcomes of the game are the three outcomes that don t occur with positive probability in equilibrium. It s just the defect defect outcome the only one that s pareto inefficient that occurs in equilibrium. One last game to look at and this is going to refresh and reinforce the point about indifference in pareto efficiency. So this is a modified version of the stag hunt.
Where player. 2 is getting a payoff of 3 for the hare hare outcome in the hare hare outcome. Despite the fact that player. 2.
Is getting a payoff of 3. That outcome is still going to be pareto inefficient. That s because we can switch to the stag stag outcome. Yes.
Player two receives same payoff so she is not being made any better off. But she s not being made any worse off whereas player. One is being made better off by switching that one that s small payoff to that larger payoff of three so hair hair. Here is pareto.
Inefficient and we re now utilizing the in difference condition in pareto efficiency. As long as one party is better off and no one is worse off. Then we have one outcome being pareto efficient or pareto dominant compared to a different outcome that wraps up this lecture on pareto efficiency. The key takeaway here again is that we can t make these interpersonal comparisons of utilities.
Instead. All we can really do is look at outcomes and compare outcomes to check to see whether we can enhance welfare for both players or at least one player without making anyone worse off. I hope you enjoyed this and hope to see. ” .
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