How to achieve the maximum social optimum

This post is more of a question than a summary. In Math Corner on Friday, we came to a cool conclusion about politics and economics, given below
POINT I: Adam Smith proposes an invisible hand guiding the market and that if we all play to our self interests we will achieve a market optimum.
COUNTERPOINT I: Our calculation form math corner shows otherwise and that if all players play selfishly, then society does not achieve its maximum social welfare. This implies that that sometimes regulation is necessary to create good market outcomes.
POINT II: Even through Prof Immorlica created some regulation by persuading us to play the (4,4) equilibrium, many of us played (1,1) because the smallest amount of selfishness would have prevented us from getting all the points
COUNTERPONT II: ??? What is the game theory counterpoint? Is regulation or deregulation better for the market? How can the rules be changed to maximize the realized social optimum?

Regarding the game, if she had been like a government and said that we are free to choose whichever we like, but if you do not go with the socially optimal solution of (4,4), you will be taxed 5 points, making the payoffs -4, -3, -2, and 4 instead of 1, 2, 3, 4, would anyone dared have been selfish?

I also wonder how many people in the game truly played a 1, versus those who physically saw a 1 being played by someone else and realized that there was no point to play anything else.

I feel like there are few ways you can make a society reach social optimum. One way easy to do it is putting a tax on people so if they don't pick the social optimum they are directly punished. I don't punishment by the government is the way to go, nor do I believe government should be the one to decide the regulations. So these are some other ways social optimum could be reached:

1. Build trust: This game is based on the trust. Most people only picked 1 because we did not trust each other not to pick 1. The lack of trust cause us all to pick the lowest socially optimum score. What we need to do is build the trust with each other that we will all will do the right thing and when we trust everyone else, then we can pick the high choice and not be afraid

2. Build sympathy and caring: This game as in most games, we only care about our own outcome. However, we as humans able to care more than just yourselves. If we care about other people outcomes and not just ourselves, we would be naturally drawn to the social optimal solution because we would want the best solution for everybody. Us being happy when other people do well and us being unhappy when other people do poorly is important in making pick the best choice for everyone.

3. Build responsibly: In this game everyone's response is anonymous. This meant there was no responsibility if you were the one to screw everyone over by picking the lowest number. However, if the game was changed so the person who screwed everyone over by picking the lowest number we brought in front of the class so we could all see them, all of a sudden there would be a sense of responsibility for people's actions and people would start to do the right thing.

All of these ways are a lot harder to do then just putting on a tax, however, I believe they are much more valuable as a society than just the punishment by the government. All of those three skills are beneficial to other aspects of life than just the game, and they all will make society better off as a whole.