Description

This video shows how repeated games work. By using the prisoner’s dilemma, we analyse how repeated games might have different results depending on whether they have a finite or infinite number of repetitions.

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In game theory, repeated games, also known as supergames, are those that play out over and over for a period of time, and therefore are usually represented using the extensive form. As opposed to one-shot games, repeated games introduce a new series of incentives: the possibility of cooperating means that we may decide to compromise in order to carry on receiving a payoff over time, knowing that if we do not uphold our end of the deal, our opponent may decide not to either. Our offer of cooperation or our threat to cease cooperation has to be credible in order for our opponent to uphold their end of the bargain. Working out whether credibility is merited simply involves working out what weighs more: the payoff we stand to gain if we break our pact at any given moment and gain an exceptional, one off payoff, or continued cooperation with lower payoffs which may or may not add up to more over a given time. Therefore, each player must consider their opponent’s possible punishment strategies.

This means that the strategy space is greater than in any regular simultaneous or sequential game. Each player will determine their strategies or moves taking into account all previous moves up until that moment. Also, since each player will take into account this information, they will play the game based on the behaviour of the opponent, and therefore must consider also possible changes in the behaviour of the latter when making choices.

Learn more by reading the dictionary entry.

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Description

This video shows how to look for a subgame perfect equilibrium. We start by explaining what subgames are, then look for a Nash equilibrium, and finally look for the subgame equilibrium.

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In game theory, a subgame is a subset of any game that includes an initial node (which has to be independent from any information set) and all its successor nodes. It’s quite easy to understand how subgames work using the extensive form when describing the game.

A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. In order to find the subgame-perfect equilibrium, we must do a backwards induction, starting at the last move of the game, then proceed to the second to last move, and so on.

Learn more by reading the dictionary entry.

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Description

This video shows how mixed strategies work. By using mixed strategies, we will be able to understand how to solve the game known as battle of the sexes.

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Mixed strategies need to be analysed in game theory when there are many possible equilibria, which is especially the case for coordination games. The battle of the sexes is a common example of a coordination game where two Nash equilibria appear, meaning that no real equilibrium can be reached.

In the battle of the sexes, a couple argues over what to do over the weekend. In this game, knowing your opponent’s strategy will not help you decide on your own course of action, and there is a chance an equilibrium may not be reached. The way to solve this dilemma is through the use of mixed strategies, in which we look at the probability of our opponent choosing one or the other strategy and balance our pay off against it.

Learn more by reading the dictionary entry.

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Description

This video explains how the battle of the sexes work. We explain how, by searching for a Nash equilibrium, we won’t be able to solve the game.

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In the battle of the sexes, a couple argues over what to do over the weekend. Both know that they want to spend the weekend together, but they cannot agree over what to do. The man prefers to go watch a boxing match, whereas the woman wants to go shopping. This is a classical example of a coordination game, analysed in game theory for its applications in many fields, such as business management or military operations.

In this case, knowing your opponent’s strategy will not help you decide on your own course of action, and there is a chance an equilibrium may not be reached. This can be easily seen by looking for a dominant strategy, eliminating all dominated strategies. However, there will be two dominant strategies, two Nash equilibria. The way to solve this dilemma is through the use of mixed strategies, in which we look at the probability of our opponent choosing one or the other strategy and balance our pay off against it.

Learn more by reading the dictionary entry.

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Description

This video explains how dominant strategies work, and how to reach a Nash equilibrium. We start by analysing dominant strategies, then explain what the Nash equilibrium is. Finally, we show an example of elimination of dominated strategies.

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Nash equilibria are defined as the combination of strategies in a game in such a way, that there is no incentive for players to deviate from their choice. This is the best option a player can make, taking into account the other players’ decision and where a change in a player’s decision will only lead to a worse result if the other players stick to their strategy. One of the best known Nash equilibria can be found in the prisoner’s dilemma. This concept belongs to game theory, specifically to non-cooperative games, and was named after John Nash who developed it.

There are a few consistency requirements that must be taken into account when dealing with Nash equilibria. One of them is known as common knowledge, which extends the necessity of complete information. Therefore, expectations about other player’s strategies must be rational.

A Nash equilibrium is therefore a combination of beliefs about probabilities over strategies and the choices of the other player.

Nash equilibria can be used to predict the outcome of finite games, whenever such equilibrium exists. Furthermore, the best equilibrium outcome can be found by using the method of elimination of dominated strategies, which will help us find the best Nash equilibrium by excluding ‘unreasonable’ Nash equilibria.

Learn more by reading the dictionary entry.

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Description

This video shows how the Battle of the Bismarck Sea works. We start by explaining the military operation, and then see all possible outcomes. Finally, we analyse the game using a game matrix.

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The Battle of the Bismarck Sea was a battle fought in February 1943 in Southeast Asia during World War II, between the Japanese Navy and the US Air Force. In game theory, its modeling was done by O. G. Haywood, Jr. in his article “Military Decision and Game Theory”, 1954. It’s a game used in game theory to analyze zero-sum games with two players.

The game, based on the actual military operation, is based on the decision General Kenney had to make. General Kenney, as Commander of the Allied Forces in the South-west Pacific Area, received intelligence reports indicating part of the Japanese Navy was about to sail from Rabaul, in the island of New Britain, to Lae, in New Guinea. Knowing this, General Kenney decided to make his five-step “Estimate of the Situation”, a technique used in US military operations.

In this game, Kenney has no dominant strategy (the sum of the payoffs of the first strategy equals the sum of the second strategy), but the Japanese do have a weakly dominating strategy, which is to go North (the payoffs are equal for one strategy but strictly better for the other). Since only one of them has a dominant strategy, there is no dominant strategy equilibrium. We must then proceed by eliminating dominated strategies. As we’ve already mentioned, for the Japanese the strategy ‘go North’ weakly dominates strategy ‘go South’. Therefore, we eliminate the strategy ‘go South’ for the Japanese, who will go North. Now that we only consider the Japanese going North, Kenney’s strategy ‘go North’ is strictly dominant over strategy ‘go South’, which will be eliminated. Therefore, North-North is the weak-dominance equilibrium.

Learn more by reading the dictionary entry.

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Description

This video explains how the prisoner’s dilemma game works. We start by explaining the prisoners’ situation, then we see how to solve the game, and analyse its consequences.

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The prisoner’s dilemma is probably the most widely used game in game theory. Its use has transcended Economics, being used in fields such as business management, psychology or biology, to name a few. Nicknamed in 1950 by Albert W. Tucker, who developed it from earlier works, it describes a situation where two prisoners, suspected of burglary, are taken into custody. However, policemen do not have enough evidence to convict them of that crime, only to convict them on the charge of possession of stolen goods.

If none of them confesses (they cooperate with each other), they will both be charged the lesser sentence, a year of prison each. The police will question them on separate interrogation rooms, which means that the two prisoners cannot communicate (hence imperfect information). The police will try to convince each prisoner to confess the crime by offering them a “get out of jail free card”, while the other prisoner will be sentenced to a ten years term. If both prisoners confess (and therefore they defect), each prisoner will be sentenced to eight years. Both prisoners are offered the same deal and know the consequences of each action (complete information) and are completely aware that the other prisoner has been offered the exact same deal (therefore, it’s common knowledge).

Since prisoners cannot communicate and will (supposedly) make their decision at the same time, this is considered to be a simultaneous game, and can be analysed using the strategic form.

Learn more by reading the dictionary entry.

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Description

This video explains what the extensive form is. We start by learning how to build a game tree to analyse games, and then use a couple of examples to see how to use it.

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In game theory, the extensive form is a way of describing a game using a game tree. It’s simply a diagram that shows that choices are made at different points in time (corresponding to each node). The payoffs are represented at the end of each branch. Since the extensive form represents decisions at different moments, it’s usually used to describe sequential games, while simultaneous games are described using the strategic form. Since sequential games imply making decisions at different moments for each player, information is perfect since each player can see the decision taken by the previous player, complete and the rules of the game and each player’s payoffs are common knowledge.

A good example of a sequential game described with the extensive form is when considering collusion agreements. Two firms share the market, colluding and maintaining high prices. Each firm can decide to stop colluding and start a price war, in order to increase their market share, even force the other to quit the market. It’s easy to see that collude-collude is both the Nash equilibrium and a Pareto optimum situation. This result may change when considering repeated games.

It’s worth mentioning that the extensive form can be used also to describe simultaneous games, by using information sets. These information sets, usually represented by a dashed line uniting two nodes or by encircling them, mean that the player does not know in which node he is, which implies imperfect information, like when using the strategic form.

Learn more by reading the dictionary entry.

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Description

This video explains what the strategic form is. We start by learning how to build a matrix to analyse games, and then use a couple of examples to see how to use it.

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In game theory, the strategic form (or normal form) is a way of describing a game using a matrix. The game is defined by exhibiting on each side of the matrix the different players, each strategy or choice they can make, and sets of payoffs they will each receive for a given strategy.

The strategic form allows us to quickly analyse each possible outcome of a game. It is usually the right description for simultaneous games, where both players choose simultaneously, as opposed to sequential games for which is better to describe the game using the extensive form (or tree form). It’s worth mentioning that simultaneous games imply there is complete and imperfect information, and the rules of the game as well as each player’s payoffs are common knowledge.

Although is not very common, sequential games can also be described using the strategic form.

Learn more by reading the dictionary entry.

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