Nash Equilibrium
A lot of this chapter is the authors' defense of Nash Equilibrium. They address a number of criticisms of the theory and provide their counter arguments.
Quick refresher... a Nash Equilibrium is when all players figure out that there is a single action each should take to maximize their return. Effectively it is the combination of actions that is everyone's ideal play in a particular game.
Why is this important?
As a game designer, I think it is crucial to know if your game has ideal plays and how easy it is for players to recognize those ideal plays. In some cases I want it to be hard for players to find a single ideal play. I want multiple options to at least seem equally viable. Otherwise people do not feel like they are making important choices.
In other cases I want players to realize what their best options are given another player's move. If it is not obvious how to counter one player's move, that can lead to a bad experience for all. This is especially important in asymmetric games.
If we assume that players can logically or intuitively find Nash Equilibrium, obstacles are important to know about. In no particular order, here are some of the obstacles they discussed. All of which can lead people to playing non-ideal strategies whether you want them to or not.
Loss Aversion
Loss aversion is the well studied psychological effect where people are much more concerned about what they might lose than what they might gain in any particular situation. The loss of $100 is felt more than a gain of $100.
Consequently, while the math may definitively point to an ideal play, loss aversion might steer people away from it.
Belief of Others' Choices
One of the cornerstones of Nash Equilibrium (and most of game theory it appears) is that each player has perfect knowledge of the other players' choices and the associated values of those choices to those players.
But if one player is wrong about their assumptions or misinformed or simply makes a mistake, they can think they have an ideal strategy when they do not. [ 1 ]
Complicated Systems
This is not a specific concern that the authors' raise but there is a part of this chapter that is very math heavy: "...each player's payoff is a quadratic function of his own strategy..." I guarantee you that very few players are doing quadratic functions in their heads at the table. Some might get close. Some might intuit an approximate answer. Some might do the equations in their spare time at home. But most will not.
Which means that a few people may know or be very close to knowing ideal plays while others will not.
Experience
As hinted at above, people often do not figure out a Nash Equilibrium immediately. Sometimes it takes trial and error before players figure out the ideal plays. [ 2 ]
But unless the same group of people have the same amount of experience, you can end up with people who have different understandings of the ideal play. In some cases this might be ok. Some games are designed to be very competitive. But this is also why cooperative games can turn into solo games.
Example Time
Pole Bidding in Championship Formula Racing is a great example of all of these barriers to finding ideal plays.
Starting a race of CFR in the front row is better than starting in the back. The game tries to balance this by making people bid for position on the starting grid.
But several players have noted to me that people generally do not bid enough to balance out the various grid positions. Statistical analysis suggests that they are correct.
Why?
I assume that there is a Nash Equilibrium that exists for every position on the grid and what the ideal bid is for all players that would result in an even valuation of those spots. However, I can see that there are a lot of reasons we don't end up there.
I think the biggest issue is complication. There are a ton of variables involved in this calculation including the number of players (from 2 to 12) and different tracks. I honestly have no idea what the value of these bids should be. Since I'm not sure anyone can accurately value these bids, no player will have perfect information about everyone's choices. Loss aversion could also be a factor. You lose what you bid... so that could certainly tamp down the amount people are willing to bid. Given the wide number of tracks available and the fact that many players I know play against a wide range of competition, experience is hard to apply from one race to another.
Solutions?
One adjustment I made to encourage larger bids was to break the bidding into two parts. The idea was to create more of an auction feel. People are generally known to over-bid in auctions because the desire to win an auction can skew one's valuation of the thing to be won... turns out my players are smarter than that.
I'm trying to solve the complication barrier by doing more and more statistical analysis. It might help eventually, but I'm not there yet.
Experience has always been my hope for the fix. That would require a bit of a closed ecosystem of players and tracks that I tend to avoid but others could certainly enforce within their own groups.
Conclusions
I hope I illustrated above how these criticisms of Nash Equilibrium can be very helpful in trying to solve game design challenges. I definitely think that the logic can be useful even if it has not solved my bidding problem yet.
In Case You Missed It
Last chapter I discussed how I thought that the tools they discuss can be tools for finding broken bits of a game design -- Nash Equilibrium included. This chapter they offered up another variant in that vein: plays that are "never the best response."
Footnotes and Sidebars
[ 1 ] Interestingly, this idea that the correct application of game theory hinges on perfect information about your opponents' choices and how they value them reminds me that there are games that can be broken by random play. I had a good friend who would play poker without looking at his cards. Which made people's attempts to determine his strategy hopeless.
Quick refresher... a Nash Equilibrium is when all players figure out that there is a single action each should take to maximize their return. Effectively it is the combination of actions that is everyone's ideal play in a particular game.
Why is this important?
As a game designer, I think it is crucial to know if your game has ideal plays and how easy it is for players to recognize those ideal plays. In some cases I want it to be hard for players to find a single ideal play. I want multiple options to at least seem equally viable. Otherwise people do not feel like they are making important choices.
In other cases I want players to realize what their best options are given another player's move. If it is not obvious how to counter one player's move, that can lead to a bad experience for all. This is especially important in asymmetric games.
If we assume that players can logically or intuitively find Nash Equilibrium, obstacles are important to know about. In no particular order, here are some of the obstacles they discussed. All of which can lead people to playing non-ideal strategies whether you want them to or not.
Loss Aversion
Loss aversion is the well studied psychological effect where people are much more concerned about what they might lose than what they might gain in any particular situation. The loss of $100 is felt more than a gain of $100.
Consequently, while the math may definitively point to an ideal play, loss aversion might steer people away from it.
Belief of Others' Choices
One of the cornerstones of Nash Equilibrium (and most of game theory it appears) is that each player has perfect knowledge of the other players' choices and the associated values of those choices to those players.
But if one player is wrong about their assumptions or misinformed or simply makes a mistake, they can think they have an ideal strategy when they do not. [ 1 ]
Complicated Systems
This is not a specific concern that the authors' raise but there is a part of this chapter that is very math heavy: "...each player's payoff is a quadratic function of his own strategy..." I guarantee you that very few players are doing quadratic functions in their heads at the table. Some might get close. Some might intuit an approximate answer. Some might do the equations in their spare time at home. But most will not.
Which means that a few people may know or be very close to knowing ideal plays while others will not.
Experience
As hinted at above, people often do not figure out a Nash Equilibrium immediately. Sometimes it takes trial and error before players figure out the ideal plays. [ 2 ]
But unless the same group of people have the same amount of experience, you can end up with people who have different understandings of the ideal play. In some cases this might be ok. Some games are designed to be very competitive. But this is also why cooperative games can turn into solo games.
Example Time
Pole Bidding in Championship Formula Racing is a great example of all of these barriers to finding ideal plays.
Starting a race of CFR in the front row is better than starting in the back. The game tries to balance this by making people bid for position on the starting grid.
But several players have noted to me that people generally do not bid enough to balance out the various grid positions. Statistical analysis suggests that they are correct.
Why?
I assume that there is a Nash Equilibrium that exists for every position on the grid and what the ideal bid is for all players that would result in an even valuation of those spots. However, I can see that there are a lot of reasons we don't end up there.
I think the biggest issue is complication. There are a ton of variables involved in this calculation including the number of players (from 2 to 12) and different tracks. I honestly have no idea what the value of these bids should be. Since I'm not sure anyone can accurately value these bids, no player will have perfect information about everyone's choices. Loss aversion could also be a factor. You lose what you bid... so that could certainly tamp down the amount people are willing to bid. Given the wide number of tracks available and the fact that many players I know play against a wide range of competition, experience is hard to apply from one race to another.
Solutions?
One adjustment I made to encourage larger bids was to break the bidding into two parts. The idea was to create more of an auction feel. People are generally known to over-bid in auctions because the desire to win an auction can skew one's valuation of the thing to be won... turns out my players are smarter than that.
I'm trying to solve the complication barrier by doing more and more statistical analysis. It might help eventually, but I'm not there yet.
Experience has always been my hope for the fix. That would require a bit of a closed ecosystem of players and tracks that I tend to avoid but others could certainly enforce within their own groups.
Conclusions
I hope I illustrated above how these criticisms of Nash Equilibrium can be very helpful in trying to solve game design challenges. I definitely think that the logic can be useful even if it has not solved my bidding problem yet.
In Case You Missed It
Last chapter I discussed how I thought that the tools they discuss can be tools for finding broken bits of a game design -- Nash Equilibrium included. This chapter they offered up another variant in that vein: plays that are "never the best response."
Footnotes and Sidebars
[ 1 ] Interestingly, this idea that the correct application of game theory hinges on perfect information about your opponents' choices and how they value them reminds me that there are games that can be broken by random play. I had a good friend who would play poker without looking at his cards. Which made people's attempts to determine his strategy hopeless.
[ 2 ] Playtesting note: the authors note that people learn somewhat faster by observing others play than by playing themselves.
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