# Subgame perfect equilibrium

Last updated
Subgame Perfect Equilibrium
A solution concept in game theory
Relationship
Subset of Nash equilibrium
Intersects with Evolutionarily stable strategy
Significance
Proposed by Reinhard Selten
Used for Extensive form games
Example Ultimatum game

In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that if the players played any smaller game that consisted of only one part of the larger game, their behavior would represent a Nash equilibrium of that smaller game. Every finite extensive game with perfect recall has a subgame perfect equilibrium. [1]

Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science, and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

## Contents

A common method for determining subgame perfect equilibria in the case of a finite game is backward induction. Here one first considers the last actions of the game and determines which actions the final mover should take in each possible circumstance to maximize his/her utility. One then supposes that the last actor will do these actions, and considers the second to last actions, again choosing those that maximize that actor's utility. This process continues until one reaches the first move of the game. The strategies which remain are the set of all subgame perfect equilibria for finite-horizon extensive games of perfect information. [1] However, backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets.

Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by first considering the last time a decision might be made and choosing what to do in any situation at that time. Using this information, one can then determine what to do at the second-to-last time of decision. This process continues backwards until one has determined the best action for every possible situation at every point in time. It was first used by Zermelo in 1913, to prove that chess has pure optimal strategies.

Within economics, the concept of utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or satisfaction within the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a consumer's preference ordering over a choice set. It is devoid of its original interpretation as a measurement of the pleasure or satisfaction obtained by the consumer from that choice.

In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed. If the game has perfect information, every information set contains only one member, namely the point actually reached at that stage of the game. Otherwise, it is the case that some players cannot be sure exactly what has taken place so far in the game and what their position is.

A subgame perfect equilibrium necessarily satisfies the one-shot deviation principle.

The one-shot deviation principle is the principle of optimality of dynamic programming applied to game theory. It says that a strategy profile of a finite extensive-form game is a subgame perfect equilibrium (SPE) if and only if there exist no profitable one-shot deviations for each subgame and every player. In simpler terms, if no player can increase their payoffs by deviating a single decision, or period, from their original strategy, then the strategy that they have chosen is a SPE. As a result, no player can profit from deviating from the strategy for one period and then reverting to the strategy.

The set of subgame perfect equilibria for a given game is always a subset of the set of Nash equilibria for that game. In some cases the sets can be identical.

The Ultimatum game provides an intuitive example of a game with fewer subgame perfect equilibria than Nash equilibria.

The ultimatum game is a game that has become a popular instrument of economic experiments. It was first described by Werner Güth, Rolf Schmittberger, and Bernd Schwarze: One player, the proposer, is endowed with a sum of money. The proposer is tasked with splitting it with another player, the responder. Once the proposer communicates their decision, the responder may accept it or reject it. If the responder accepts, the money is split per the proposal; if the responder rejects, both players receive nothing. Both players know in advance the consequences of the responder accepting or rejecting the offer.

## Example

Determining the subgame perfect equilibrium by using backward induction is shown below in Figure 1. Strategies for Player 1 are given by {Up, Uq, Dp, Dq}, whereas Player 2 has the strategies among {TL, TR, BL, BR}. There are 4 subgames in this example, with 3 proper subgames.

Using the backward induction, the players will take the following actions for each subgame:

• Subgame for actions p and q: Player 1 will take action p with payoff (3, 3) to maximize Player 1's payoff, so the payoff for action L becomes (3,3).
• Subgame for actions L and R: Player 2 will take action L for 3 > 2, so the payoff for action D becomes (3, 3).
• Subgame for actions T and B: Player 2 will take action T to maximize Player 2's payoff, so the payoff for action U becomes (1, 4).
• Subgame for actions U and D: Player 1 will take action D to maximize Player 1's payoff.

Thus, the subgame perfect equilibrium is {Dp, TL} with the payoff (3, 3).

An extensive-form game with incomplete information is presented below in Figure 2. Note that the node for Player 1 with actions A and B, and all succeeding actions is a subgame. Player 2's nodes are not a subgame as they are part of the same information set.

The first normal-form game is the normal form representation of the whole extensive-form game. Based on the provided information, (UA, X), (DA, Y), and (DB, Y) are all Nash equilibria for the entire game.

The second normal-form game is the normal form representation of the subgame starting from Player 1's second node with actions A and B. For the second normal-form game, the Nash equilibrium of the subgame is (A, X).

For the entire game Nash equilibria (DA, Y) and (DB, Y) are not subgame perfect equlibria because the move of Player 2 does not constitute a Nash Equilibrium. The Nash equilibrium (UA, X) is subgame perfect because it incorporates the subgame Nash equilibrium (A, X) as part of its strategy. [2]

To solve this game, first find the Nash Equilibria by mutual best response of Subgame 1. Then use backwards induction and plug in (A,X) → (3,4) so that (3,4) become the payoffs for Subgame 2. [2]

The dashed line indicates that player 2 does not know whether player 1 will play A or B in a simultaneous game.

Player 1 chooses U rather than D because 3 > 2 for Player 1's payoff. The resulting equilibrium is (A, X) → (3,4).

Thus, the subgame perfect equilibrium through backwards induction is (UA, X) with the payoff (3, 4).

## Subgame-perfect equilibrium in finitely repeated games

For finitely repeated games, if a stage game has only one unique Nash equilibrium, the subgame perfect equilibrium is to play without considering past actions, treating the current subgame as a one-shot game. An example of this is a finitely repeated Prisoner's dilemma game. Using backward induction, the last subgame in a finitely repeated Prisoner's dilemma requires players to play the unique Nash equilibrium (both players defecting). Because of this, all games prior to the last subgame will also play the Nash equilibrium to maximize their single-period payoffs.

If a stage-game in a finitely repeated game has multiple Nash equilibria, subgame perfect equilibria can be constructed to play non-stage-game Nash equilibrium actions, through a “carrot and stick” structure. One player can use the one stage-game Nash equilibrium to incentivize playing the non-Nash equilibrium action, while using a stage-game Nash equilibrium with lower payoff to the other player if they choose to defect. [3]

## Finding subgame-perfect equilibria

Reinhard Selten proved that any game which can be broken into "sub-games" containing a sub-set of all the available choices in the main game will have a subgame perfect Nash Equilibrium strategy (possibly as a mixed strategy giving non-deterministic sub-game decisions). Subgame perfection is only used with games of complete information. Subgame perfection can be used with extensive form games of complete but imperfect information.

The subgame-perfect Nash equilibrium is normally deduced by "backward induction" from the various ultimate outcomes of the game, eliminating branches which would involve any player making a move that is not credible (because it is not optimal) from that node. One game in which the backward induction solution is well known is tic-tac-toe, but in theory even Go has such an optimum strategy for all players.

The interesting aspect of the word "credible" in the preceding paragraph is that taken as a whole (disregarding the irreversibility of reaching sub-games) strategies exist which are superior to subgame perfect strategies, but which are not credible in the sense that a threat to carry them out will harm the player making the threat and prevent that combination of strategies. For instance in the game of "chicken" if one player has the option of ripping the steering wheel from their car they should always take it because it leads to a "sub game" in which their rational opponent is precluded from doing the same thing (and killing them both). The wheel-ripper will always win the game (making his opponent swerve away), and the opponent's threat to suicidally follow suit is not credible.

## Related Research Articles

In game theory, the centipede game, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.

In game theory, a subgame is any part of a game that meets the following criteria :

1. It has a single initial node that is the only member of that node's information set.
2. If a node is contained in the subgame then so are all of its successors.
3. If a node in a particular information set is in the subgame then all members of that information set belong to the subgame.

In game theory, a player's strategy is any of the options which he or she chooses in a setting where the outcome depends not only on their own actions but on the actions of others. A player's strategy will determine the action which the player will take at any stage of the game.

An extensive-form game is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature".

In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information. A PBE is a refinement of both Bayesian Nash equilibrium (BNE) and subgame perfect equilibrium (SPE). A PBE has two components - strategies and beliefs:

In game theory, a Bayesian game is a game in which players have incomplete information about the other players. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. These beliefs are represented by a probability distribution over the possible payoff functions.

In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. Importantly, the later players must have some information of the first's choice, otherwise the difference in time would have no strategic effect. Sequential games hence are governed by the time axis, and represented in the form of decision trees.

In game theory, trembling hand perfect equilibrium is a refinement of Nash equilibrium due to Reinhard Selten. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability.

In game theory, folk theorems are a class of theorems about possible Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept subgame-perfect Nash equilibria rather than Nash equilibrium.

In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. Single stage game or single shot game are names for non-repeated games.

Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. A sequential equilibrium specifies not only a strategy for each of the players but also a belief for each of the players. A belief gives, for each information set of the game belonging to the player, a probability distribution on the nodes in the information set. A profile of strategies and beliefs is called an assessment for the game. Informally speaking, an assessment is a perfect Bayesian equilibrium if its strategies are sensible given its beliefs and its beliefs are confirmed on the outcome path given by its strategies. The definition of sequential equilibrium further requires that there be arbitrarily small perturbations of beliefs and associated strategies with the same property.

In game theory, a Manipulated Nash equilibrium or MAPNASH is a refinement of subgame perfect equilibrium used in dynamic games of imperfect information. Informally, a strategy set is a MAPNASH of a game if it would be a subgame perfect equilibrium of the game if the game had perfect information. MAPNASH were first suggested by Amershi, Sadanand, and Sadanand (1988) and has been discussed in several papers since. It is a solution concept based on how players think about other players' thought processes.

Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

A non-credible threat is a term used in game theory and economics to describe a threat in a sequential game that a rational player would actually not carry out, because it would not be in his best interest to do so.

A Markov perfect equilibrium is an equilibrium concept in game theory. It is the refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be readily identified. The term appeared in publications starting about 1988 in the work of economists Jean Tirole and Eric Maskin. It has since been used, among else, in the analysis of industrial organization, macroeconomics and political economy.

Mertens stability is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens. This solution concept is now called Mertens stability, or just stability.

## References

1. Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.
2. Joel., Watson. Strategy : an introduction to game theory (Third ed.). New York. ISBN   9780393918380. OCLC   842323069.
3. Takako, Fujiwara-Greve. Non-cooperative game theory. Tokyo. ISBN   9784431556442. OCLC   911616270.