What are some examples of real-world bargaining games that involve mixed strategies?

Imagine you and a friend have a big bag of candies. You both want to have more candies, but you also want to play fair. Now, think about playing a game where sometimes you will give your friend some candies and sometimes you take some candies from them. You both are trying to figure out the best ways to share the candies by sometimes giving, sometimes taking, and sometimes trading candies. This is called a "bargaining game". And the mixed strategy is like deciding to trade a gummy bear for a chocolate sometimes but not always, to make the game more exciting and fun. It means you have many different choices and you choose them in different ways each time you play. It's a way to keep the game surprising and fair for both of you!

How parties negotiate contracts. Here is an article on the topic: https://digitalcommons.pepperdine.edu/jbel/vol4/iss1/1/

Basically because it convexifies the strategy space allowing one to show existence of equilibria. This was Nash's great insight.

Think of a big marketplace where many people come to buy and sell things. Let's say there is a seller who has a shop of beautiful lamps. Now, this seller doesn't have a fixed price on these lamps. Some days he asks for a high price, and some days he goes for a lower price. He mixes up his pricing strategy to attract different buyers - some who are okay with paying more and others who are looking for a bargain. This is a mixed strategy because the seller is not sticking to one single plan; he changes it based on who is buying and what day it is, trying to sell as many lamps as possible. It's like sometimes trading a bigger toy for two smaller toys and other times not trading at all to see what the best deal you can get is!

These examples might be a little bit misleading. Ssince you can't do half of one thing and half of the other thing at the same time, mixing strategies is about the probability, not the action. The last example--deciding who to vote for based on a coin flip or polls--is much better than the cake example, where the emphasis should be on the "random" decision rather than the actual strategy of cutting the cake and letting the other person decide.

There is a very interesting use of mixed strategies and randomization in nuclear strategy. Both sides know that the actual use of nuclear weapons is irrational, because of mutually assured destruction. So, in order to make your threat of using them credible, you effectively randomize your use of the weapons by partly taking away your ability to fully control their use. This might be done by delegating to multiple people the unilateral authority to use them, putting those with access to the weapons on "high alert", or reducing checks on their use etc. If the counterparty knows that their use could be triggered on a hairpin by a mistaken radar blip, the threat is more serious. It is part of what makes nuclear weapons so terrifying.

While those ignorant of the topic would be impressed, people with expertise on the topic would find it highly unsatisfactory. Here is a quote from professional game theorist Ariel Rubinstein: Nearly every book on game theory begins with the sentence: ‘Game theory is relevant to …’ and is followed by an endless list of fields, such as nuclear strategy, financial markets, the world of butterflies and flowers, and intimate situations between men and women. Articles citing game theory as a source for resolving the world’s problems are frequently published in the daily press. But after nearly forty years of engaging in this field, I have yet to find even a single application of game theory in my daily life”

Here are some I can think of that the data science community may find interesting: Project Bidding – Data scientists might use a mix of strategies such as lowering their service fees or offering additional services to win project contracts. Algorithm Tuning – While optimizing an algorithm, they might adopt mixed strategies, sometimes focusing on improving speed while other times focusing on accuracy. Resource Allocation – When managing a team, a data scientist might alternate between assigning more people to critical projects and evenly distributing resources across projects based on priority and deadline changes.