Sheldon from Big Bang Theory states that if someone gives you Christmas gifts, the “essence of the custom” is that I must buy you something comparable in value, showing the same level of friendship.

The mathematics might help you decide who and what gifts to buy for Christmas.

Consider two people (players 1, 2 and 3) who must decide if they will give each other gifts. There are two choices or strategies that each person can choose from. They have two options. If they get a gift, they feel the “E”. They may have to pay a cost (“C”) if they gift a present. This can be either financial or based upon effort.

The payoff matrix represents the possible outcomes of this game. Each entry consists of two parts: The payoff for player 1 (payoff of player 2) and the payoff to player 2.

Both players can purchase gifts for each other. The reward is both the enjoyment and the cost of the gift. The payoff for a player who buys a gift and not the other is the enjoyment of the present. The person who doesn’t buy a present gets nothing but enjoyment. The payoff for each person is zero if neither of them buys a gift.

This game encourages cooperation through giving gifts to one another, which seems counterintuitive. E-C is, in our example, a positive return on investment. If one player is confident that the other will cooperate, they might prefer not to purchase a gift. They will both get the gift at no cost. If they both fail to do so, they will each lose.

Math tells us that it is better not to exchange gifts for the game. This is the Nah equilibrium. It means that we cannot improve our situation by changing our behavior.

You were my Christmas gift.

This assumes that we play the game only once. The Iterated Prisoner’s Dilemma is our challenge. We will share many Christmases if we play against each other.

In 1984, Robert Axelrod published The Evolution of Cooperation. It dealt with the Iterated Prisoner’s Dilemma. His research revealed that generous strategies work better than greedy strategies over the long term. Anatol Rapaport, also known as “titfortat”, proposed the winning strategy in this iterated game.

This strategy requires you to work together the first year, then follow up with what your opponent did the previous year. This could be giving a friend a gift and seeing what they do next year.

Sometimes people are unable to decide what they would like. This can lead to disappointments if gifts are not well received. Even if the gift isn’t expensive, small children can find joy in receiving their gifts. Teenagers may have a harder time because most of what they desire is more expensive, and they may enjoy getting them. However, the joy of receiving them is diminished – even if it is smaller.

Socks are always difficult to find because they’re relatively expensive, but people don’t necessarily like them. If purchased from the right retailer, vouchers can be a better choice. They have a much greater “value”.

What’s the joy in giving?

The bonus of “J” can make the game completely different. Buy gifts if the reward you get when you buy presents for someone changes from -C to J-C and is positive.

How did Sheldon solve the problem? He bought several gift baskets of different value and gave Penny his closest. Penny gave Sheldon a napkin with “the DNA Leonard Nimoy” on it. Although the gift was not of monetary value, Sheldon put enough value on it to give all his gift baskets.

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