Playing the Probabilities
Our world is chock full of game theory situations. Some of them we play on a conscious level and others we play on a subconscious level. Still others we are unaware of. In some situations, we would actually benefit from taking more time to calculate things.
Say you have a diamond that a thief is trying to steal. You possess two safes, one at work and one at home. And say the thief only has time to go to one of your safes and will only try for one evening. He has a 90% chance of cracking the safe at home and a 50% chance of cracking the safe at work. You and the thief both know this information. Which safe do you hide your diamond in?
Many would choose the safe route and choose the one at work since the thief has a 50% chance of cracking it. 50% is better than 90% right? In truth, you can do better than those odds. After all, if the thief chose work and you chose home, he would have a 0% chance of getting the diamond. Of course, if the thief thought you would purposely put it in the weaker of the safes, he might try for it there and now he has a 90% chance. You can second-guess each other all day like the Sicilian from Princess Bride, but this is never-ending speculation. Second-guessing each other is likely to go your way 50% of the time. This means the thief is likely to get your diamond 35% the time (0% averaged by the average of 50% and 90%)
You can still do better. The best system is to randomize your choices to an optimal level. If 64% of the time, you put the diamond in the better safe, and 36% of the time, you put the diamond in the worse safe, the thief’s odds of getting the diamond drop to 32% no matter which safe he chooses. Even if the thief finds out that you have randomized your choices, it is in his best interest to randomize his response and flip a coin to decide where to go. The nicest aspect of the randomization is that you do not have to regret anything. If the thief gets your diamond, you say, “oh, well, that’s my luck” and you do not have to obsess about which safe you should have chosen.
So, how does this relate to non-diamond thieves?
Say you are trying to decide whether you should marry someone and they are trying to decide if they should marry you. If you both choose “marry”, the relationship will continue. If one chooses “marry” and other person doesn’t, the relationship will end. If you both choose not to marry, the relationship will also continue.
So let’s add some math to this. Let’s say that Person A is slightly happier unmarried and Person B is slightly happier married. Both are much worse off broken up. What should Person A and Person B do if the marriage question arises?
................................................Person A
.........................................Marry....... Not Marry
Person B.......... Marry..... 40, 80...... 20, 20
.................Not Marry..... 20, 20....... 90, 30
Now, if we look at the happiness levels above, we find that if both choose “marry”, Person A has a 40% chance at happiness while Person B has a 80% chance. If both choose “not marry”, Person A has a 90% chance of happiness while Person B has a 30% chance. If they mismatch their choices, they break up and both only have a 20% chance at happiness.
Now, in this situation, if either person leads, the other will have to reluctantly follow. If Person A declares, “I don’t want to get married,” Person B will respond, “I don’t either” (since 30% is still more than 20%). Conversely, if Person B declares, “I want to get married,” Person A will respond, “I do too” (since 40% is still greater than 20%). So, whoever goes first wins.
Realizing this, though, both would want to declare their intentions first which would result in simultaneous declarations. Simultaneous declarations, though, would lead to other choices. If Person B spat out “marry”, they would have a 50% chance at happiness (80% + 20% by 2), but they also know that Person A has a 55% percent chance of happiness spitting out “not marry”. Would “not marry” be a better choice? What if they plan on saying “marry” thinking I will say it? As you see, this would to lead to second-guessing again.
If Person B picked a word on a 50/50 split (second-guessing Person A), they would have 37.5% chance (80% + 20% + 30% + 20% divided by 4) at happiness assuming that Person A also second-guessed Person B. Person A would have a 42.5% (40% + 20% + 90% + 20% divided by 4) chance or happiness. Second-guessing is this case actually has better odds than following the other person’s lead.
Of course, optimal randomization is best the solution. If Person A randomized their choices and chose “marry” 65% of the time and “not marry” 35% of the time, they would have over a 45% chance of happiness (.65 * .55 + .35 *. 30). Now, if Person B also randomized, things would get more complicated, especially if they knew that they are both randomizing. If they did, calculations would shift and eventually settle with both parties choosing something close to an even split of saying “marry” versus “not marry” (the final resting place would favor both parties saying “marry” slightly since Person A is proportionally happiest there).
So, what has all this horrid math taught us? Leading can be very advantageous. Additionally, the fear of following a choice can force a person to choose faster since following can be an inferior position. (Who wants to be stuck in a position where someone is asking “where is this going?”) Additionally, random action and especially weighted random action can lead to optimal situations. Most importantly, weighted random action frees human beings from obsession and regret.
Thank goodness no one lives in the world of Person A and Person B, right?
Say you have a diamond that a thief is trying to steal. You possess two safes, one at work and one at home. And say the thief only has time to go to one of your safes and will only try for one evening. He has a 90% chance of cracking the safe at home and a 50% chance of cracking the safe at work. You and the thief both know this information. Which safe do you hide your diamond in?
Many would choose the safe route and choose the one at work since the thief has a 50% chance of cracking it. 50% is better than 90% right? In truth, you can do better than those odds. After all, if the thief chose work and you chose home, he would have a 0% chance of getting the diamond. Of course, if the thief thought you would purposely put it in the weaker of the safes, he might try for it there and now he has a 90% chance. You can second-guess each other all day like the Sicilian from Princess Bride, but this is never-ending speculation. Second-guessing each other is likely to go your way 50% of the time. This means the thief is likely to get your diamond 35% the time (0% averaged by the average of 50% and 90%)
You can still do better. The best system is to randomize your choices to an optimal level. If 64% of the time, you put the diamond in the better safe, and 36% of the time, you put the diamond in the worse safe, the thief’s odds of getting the diamond drop to 32% no matter which safe he chooses. Even if the thief finds out that you have randomized your choices, it is in his best interest to randomize his response and flip a coin to decide where to go. The nicest aspect of the randomization is that you do not have to regret anything. If the thief gets your diamond, you say, “oh, well, that’s my luck” and you do not have to obsess about which safe you should have chosen.
So, how does this relate to non-diamond thieves?
Say you are trying to decide whether you should marry someone and they are trying to decide if they should marry you. If you both choose “marry”, the relationship will continue. If one chooses “marry” and other person doesn’t, the relationship will end. If you both choose not to marry, the relationship will also continue.
So let’s add some math to this. Let’s say that Person A is slightly happier unmarried and Person B is slightly happier married. Both are much worse off broken up. What should Person A and Person B do if the marriage question arises?
................................................Person A
.........................................Marry....... Not Marry
Person B.......... Marry..... 40, 80...... 20, 20
.................Not Marry..... 20, 20....... 90, 30
Now, if we look at the happiness levels above, we find that if both choose “marry”, Person A has a 40% chance at happiness while Person B has a 80% chance. If both choose “not marry”, Person A has a 90% chance of happiness while Person B has a 30% chance. If they mismatch their choices, they break up and both only have a 20% chance at happiness.
Now, in this situation, if either person leads, the other will have to reluctantly follow. If Person A declares, “I don’t want to get married,” Person B will respond, “I don’t either” (since 30% is still more than 20%). Conversely, if Person B declares, “I want to get married,” Person A will respond, “I do too” (since 40% is still greater than 20%). So, whoever goes first wins.
Realizing this, though, both would want to declare their intentions first which would result in simultaneous declarations. Simultaneous declarations, though, would lead to other choices. If Person B spat out “marry”, they would have a 50% chance at happiness (80% + 20% by 2), but they also know that Person A has a 55% percent chance of happiness spitting out “not marry”. Would “not marry” be a better choice? What if they plan on saying “marry” thinking I will say it? As you see, this would to lead to second-guessing again.
If Person B picked a word on a 50/50 split (second-guessing Person A), they would have 37.5% chance (80% + 20% + 30% + 20% divided by 4) at happiness assuming that Person A also second-guessed Person B. Person A would have a 42.5% (40% + 20% + 90% + 20% divided by 4) chance or happiness. Second-guessing is this case actually has better odds than following the other person’s lead.
Of course, optimal randomization is best the solution. If Person A randomized their choices and chose “marry” 65% of the time and “not marry” 35% of the time, they would have over a 45% chance of happiness (.65 * .55 + .35 *. 30). Now, if Person B also randomized, things would get more complicated, especially if they knew that they are both randomizing. If they did, calculations would shift and eventually settle with both parties choosing something close to an even split of saying “marry” versus “not marry” (the final resting place would favor both parties saying “marry” slightly since Person A is proportionally happiest there).
So, what has all this horrid math taught us? Leading can be very advantageous. Additionally, the fear of following a choice can force a person to choose faster since following can be an inferior position. (Who wants to be stuck in a position where someone is asking “where is this going?”) Additionally, random action and especially weighted random action can lead to optimal situations. Most importantly, weighted random action frees human beings from obsession and regret.
Thank goodness no one lives in the world of Person A and Person B, right?
posted by American Dilettante at
5:46 PM
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