French0

Normally, this is where the story would end,but a wizard happened to be watching the whole mess unfold. He tells Crispy and Chewy that, as punishment for betraying each other, they’re doomed to repeat this dilemma for the rest of their lives,starting with all four limbs at each sunrise. Now what happens?This is called an Infinite Prisoner’s Dilemma, and it’s a literal game changer. That’s because the gingerbread men can now use their future decisions as bargaining chips for the present ones. Consider this strategy: both agree to spare each other every day.If one ever chooses to sacrifice, the other will retaliate by choosing “sacrifice” for the rest of eternity.

This analysis isn’t unique to cookies and wizards;we see it play out in real-life situations like trade negotiations and international politics. Rational leaders must assume that the decisions they make today will impact those of their adversaries tomorrow. Selfishness may win out in the short-term,but with the proper incentives,peaceful cooperation is not only possible, but demonstrably and mathematically ideal.As for the gingerbread men, their eternity may be pretty crumby, but so long as they go out on a limb,their friendship will never again be half-baked.

But as delta approaches 1, they’ll do anything possible to avoid the pain of infinite triple limb consumption, which means they’ll choose to spare each other.At some point in between they could go either way. We can find out where that point is by writing the infinite series that represents each strategy, setting them equal to each other,and solving for delta.That yields 1/3, meaning that as long as Crispy and Chewy care about tomorrow at least 1/3 as much as today, it’s optimal for them to spare and cooperate forever.

So is that enough to get these poor sentient baked goods to agree to cooperate?To figure that out, we have to factor in another consideration:the gingerbread men probably care about the future less than they care about the present.In other words, they might discount how much they care about their future limbs by some number,which we’ll call delta. This is similar to the idea of inflation eroding the value of money. If delta is one half,on day one they care about day 2 limbs half as much as day 1 limbs, day 3 limbs 1 quarter as much as day 1 limbs, and so on. A delta of 0 means that they don’t care about their future limbs at all, so they’ll repeat their initial choice of mutual sacrifice endlessly.

Normally, this is where the story would end,but a wizard happened to be watching the whole mess unfold. He tells Crispy and Chewy that, as punishment for betraying each other, they’re doomed to repeat this dilemma for the rest of their lives,starting with all four limbs at each sunrise. Now what happens?This is called an Infinite Prisoner’s Dilemma, and it’s a literal game changer. That’s because the gingerbread men can now use their future decisions as bargaining chips for the present ones. Consider this strategy: both agree to spare each other every day.If one ever chooses to sacrifice, the other will retaliate by choosing “sacrifice” for the rest of eternity.

First, let’s consider Chewy’s options.If Crispy spares him, Chewy can run away scot-free by sacrificing Crispy.But if Crispy sacrifices him, Chewy can keep one of his limbs if he also sacrifices Crispy.No matter what Crispy decides, Chewy always experiences the best outcome by choosing to sacrifice his companion.The same is true for Crispy. This is the standard conclusion of the Prisoner's Dilemma:the two characters will betray one another. Their strategy to unconditionally sacrifice their companion is what game theorists call the “Nash Equilibrium, meaning that neither can gain by deviating from it.Crispy and Chewy act accordingly and the smug fox runs off with a belly full of gingerbread,leaving the two former friends with just one leg to stand on.

Finally, if both choose to sacrifice,the fox will eat 3 limbs from each.In game theory, this scenario is called the “Prisoner's Dilemma.” To figure out how these gingerbread men will act in their perfect rationality,we can map the outcomes of each decision. The rows represent Crispy’s choices,and the columns are Chewy’s.Meanwhile, the numbers in each cell represent the outcomes of their decisions, as measured in the number of limbs each would keep:So do we expect their friendship to last the game?

Two perfectly rational gingerbread men,Crispy and Chewy,are out strolling when they’re caught by a fox.Seeing how happy they are, he decides that,instead of simply eating them,he’ll put their friendship to the test with a cruel dilemma.He’ll ask each gingerbread man whether he’d opt to Spare or Sacrifice the other.They can discuss,but neither will know what the otherchose until their decisions are locked in. If both choose to spare the other, the fox will eat just one of each of their limbs;if one chooses to spare while the other sacrifices,the sparer will be fully eaten, while the traitor will run away with all his limbs intact.

All of these fallibilities of memory can have real-world impacts.If police interrogations use leading questions with eye witnesses or suspects,suggestibility could result in incorrect identifications or unreliable confessions.Even in the absence of leading questions, misattribution can lead to inaccurate eyewitness testimony.In a courtroom,if a judge rules a piece of evidence inadmissible and tells jurors to disregard it,they may not be able to do so.In a medical setting, if a patient seeks a second opinion and the second physician is aware of the first one’s diagnosis,that knowledge may bias their conclusion.Our memories are not ironclad representations of reality,but subjective perceptions.And there’s not necessarily anything wrong with that—he problems arise when we treat memory as fact,rather than accepting this fundamental truth about the nature of our recollections.

In another study,researchers gave two groups of participants background information on a historical war and asked them to rate the likelihood that each side would win.They gave each group the same information,except that they only told one group who had actually won the war—the other group didn’t know the real world outcome.In theory, both groups’ answers should be similar,because the likelihood of each side winning isn’t effected by who actually won—if there’s a 20% chance of thunderstorms,and a thunderstorm happens,the chance of thunderstorms doesn’t retroactively go up to 100%.Still, the group that knew how the war ended rated the winning side as more likely to win than the group who did not.

Another study, where a psychologist questioned over 2,000 people on their views about the legalization of marijuana,highlights yet another kind of influence on memory.Participants answered questions in 1973 and 1982.Those who said they had supported marijuana legalization in 1973,but reported they were against it in 1982,were more likely to recall that they were 。actually against legalization in 1973—bringing their old views in line with their current ones.Our current opinions,feelings, and experiences can bias our memories of how we felt in the past.

In another experiment, people were shown an image of a magnifying glass,and then told to imagine a lollipop.They frequently recalled that they saw the magnifying glass and the lollipop.They struggled to link the objects to the correct context—whether they actually saw them,or simply imagined them.

Take another study,in which researchers briefly showed random collection of photographs to a group of participants,including images of a university campus none of them had ever visited.When shown the images three weeks later,a majority of participants said that they had probably or definitely visited the campus in the past.The participants misattributed information from one context— an image they’d seen—onto another— a memory of something they believed they actually experienced.

Our memories are sometimes unreliable.And though we still don’t know precisely what causes this fallibility，on a neurological level,research has highlighted some of the most common ways our memories diverge from what actually happened.The mall study highlights how we can incorporate information from outside sources,like other people or the news,into our personal recollections without realizing it.This kind of suggestibility is just one influence on our memories.

In a study in the 1990s,participants recalled getting lost in a shopping mall as children.Some shared these memories in vivid detail one even remembered that the old man who rescued him was wearing a flannel shirt.But none of these people had actually gotten lost in a mall.They produced these false memories。when the psychologists conducting the study told them they’d gotten lost,and although they might not remember the incident,their parents had confirmed it.And it wasn’t just one or two people who thought they remembered getting lost—a quarter of the participants did.These findings may sound unbelievable,but they actually reflect a very common experience.

So, think of a challenge you want to take on,realize it's not going to be easy,accept that you'll make mistakes,and be kind to yourself when you do. Give yourself a pep talk, stand up, and go for it. The excitement you'll feel knowing that whatever the result,you'll have gained greater knowledge and understanding. This is confidence.

Tip three: practice failure. Face it, you're going to fail sometimes. Everyone does.J.K. Rowling was rejected by twelve different publishers before one picked up Harry Potter. The Wright Brothers built on history's failed attempts at flight,including some of their own,before designing a successful airplane. Studies show that those who fail regularly and keep trying anyway are better equipped to respond to challenges and setbacks in a constructive way. They learn how to try different strategies,ask others for advice,and perservere.

If you have a fixed mindset,meaning that you think your talents are locked in place,you might give up,assuming you've discovered something you're not very good at. But if you have a growth mindset and think your abilities can improve,a challenge is an opportunity to learn and grow. Neuroscience supports the growth mindset. The connections in your brain do get stronger and grow with study and practice.It also turns out, on average,people who have a growth mindset are more successful,getting better grades,and doing better in the face of challenges.

Tip two: believe in your ability to improve. If you're looking for a long-term change,consider the way you think about your abilities and talents. Do you think they are fixed at birth,or that they can be developed,like a muscle? These beliefs matter because they can influence how you act when you're faced with setbacks.

Tip 1: a quick fix. There are a few tricks that can give you an immediate confidence boost in the short term. Picture your success when you're beginning a difficult task,something as simple as listening to music with deep bass;it can promote feelings of power. You can even strike a powerful pose or give yourself a pep talk.

So where does confidence even come from?There are several factors that impact confidence.One: what you're born with,such as your genes,which will impact things like the balance of neurochemicals in your brain.Two: how you're treated.This includes the social pressures of your environment.And three: the part you have control over,the choices you make,the risks you take,and how you think about and respond to challenges and setbacks.It isn't possible to completely untangle these three factors,but the personal choices we make certainly play a major role in confidence development.So, by keeping in mind a few practical tips,we do actually have the power to cultivate our own confidence.

When faced with a big challengewhere potential failure seems to lurk at every corner,maybe you've heard this advice before:Be more confident.And most likely, this is what you thinkwhen you hear it:If only it were that simple.But what is confidence?Take the belief that you are valuable,worthwhile, and capable,also known as self-esteem,add in the optimism that comes when you are certain of your abilities,and then empowered by these,act courageously to face a challenge head-on.This is confidence.It turns thoughts into action.