Game Theory 101 (#4): Pure Strategy Nash Equilibrium and the Stag Hunt

Introduction to Stag Hunt and Pure Strategy Nash Equilibrium

In this section, William Spaniel introduces the concept of the stag hunt and pure strategy Nash equilibrium. He explains the scenario of two hunters choosing between hunting a hare or a stag, and the different payoffs associated with each choice.

The Stag Hunt Scenario

• Two hunters are going out to catch meat.

• There are two hares and one stag in the range.

• The hunters can only bring equipment for one type of animal.

• They have to choose their equipment without knowing what the other hunter will choose.

• The stag is worth more (6 units of meat) than both hares combined (2 units of meat each).

• Both hunters need to choose stag hunting equipment to successfully capture the stag.

Payoff Matrix

• Player 1 and Player 2 can choose to hunt a stag or a hare.

• If both players choose a stag, they coordinate and capture 6 units of meat, splitting it evenly between them (3 units each).

• If one player chooses a stag and the other chooses a hare, the player hunting the hare gets 0 units of meat while the other player captures both hares (2 units of meat).

• If both players choose to hunt hares, they split the two hares evenly (1 hare each).

Nash Equilibrium

• Nash equilibrium is a set of strategies where no player has an incentive to change their strategy.

• We only care about individual deviations, not group deviations.

• A Nash equilibrium is stable because once strategies are revealed, there are no regrets or opportunities for individual players to improve their outcomes.

Finding Nash Equilibria

Stag-Stag Outcome

• Player 1 gets 3 units by hunting a stag and only 2 by hunting a hare, so he is satisfied with his strategy.

• Player 2 also gets 3 units by hunting a stag and only 2 by hunting a hare, so she is happy maintaining her stag strategy.

• The stag-stag outcome is a Nash equilibrium.

Other Outcomes

• The other outcomes are not Nash equilibria because at least one player would have an incentive to change their strategy for better payoffs.

Further analysis and strategies may be required to determine sensible outcomes in this game.

New Section

In this section, the speaker discusses the concept of Nash equilibria in game theory and explores the possibility of multiple equilibria.

Exploring Multiple Nash Equilibria

• The stability of a strategy depends on whether it is the best choice for both players.

• Games can have more than one Nash equilibrium, so it is important to check for additional equilibria.

New Section

The speaker analyzes a specific outcome where player 1 hunts a hare and player 2 hunts a stag to determine if it is a Nash equilibrium.

Analyzing Player 1's Choice

• If player 1 knows that player 2 is hunting a stag, he would want to change from hunting a hare to hunting a stag.

• This indicates that the outcome of player 1 hunting a hare and player 2 hunting a stag is not inherently stable as there exists an individual deviation that benefits player 1.

New Section

The speaker considers both players' potential deviations from their strategies in the previous outcome to determine if it is a Nash equilibrium.

Profitable Deviations for Both Players

• Player 2 also has a profitable deviation in this case by switching from hunting a stag (earning zero) to hunting a hare (earning one).

• Although considering player 2's deviation may seem redundant, it reinforces the fact that this outcome cannot be a Nash equilibrium due to individual deviations benefiting both players.

New Section

The speaker examines another outcome where player 1 hunts a stag and player 2 hunts a hare to determine its status as a Nash equilibrium.

Identical Situation as Previous Outcome

• The situation where player 1 hunts a stag and player 2 hunts a hare is essentially the same as the previous outcome.

• Player 1 would want to change his strategy from hunting a stag (earning zero) to hunting a hare (earning one), indicating that this outcome is not a Nash equilibrium.

New Section

The speaker analyzes the outcome where both players hunt hares to determine if it is a Nash equilibrium.

No Profitable Deviations

• Neither player has a profitable deviation in this case.

• Player 1 cannot switch from hunting a hare to hunting a stag as it would result in earning zero instead of one.

• Similarly, player 2 would be unhappy if she switched her strategy from hunting a hare, earning one, to any other option.

New Section

The speaker summarizes the findings and discusses the counterintuitive nature of Nash equilibria.

Two Nash Equilibria Found

• Two Nash equilibria are identified: when both players hunt a stag and when both players hunt a hare.

• The latter outcome may seem inefficient compared to both choosing stag, but it arises due to coordination difficulties and expectations set by external factors.

New Section

The speaker concludes by highlighting that Nash equilibria are inherently stable and do not involve regrets for either player.

Inherent Stability of Nash Equilibria

• While not always efficient or optimal, Nash equilibria are inherently stable.

• Players do not have regrets about their choices within these equilibria because they expect others to make similar decisions.