Economics
Maximin Strategy
The maximin strategy is a decision-making approach that focuses on maximizing the minimum possible outcome. In economics, it is often used in game theory and decision theory to minimize potential losses and ensure a level of security in uncertain situations. This strategy involves selecting the option that provides the highest payoff under the worst-case scenario.
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9 Key excerpts on "Maximin Strategy"
eBook - PDFInternational Negotiation
Process and Strategies
- Ho-Won Jeong(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
The real-world examples are illustrated by minimization of the risk of terrorist infiltration inside national borders and a favorable division of goods or other objects with a limited supply. A minimax procedure serves as the concept of a solution for two-person zero-sum games where one player’s utility can be improved only by making the other player worse off. A minimax solution methodology is characterized by maximization of the worst-case payoff that can occur. The minimax value is obtained by striving to minimize one’s maximum losses in anticipation of the worst scenario. The best strategy to be prepared for the worst possible consequences is to “choose the action that leads to the least bad outcomes” (Dixit and Skeath 2004 , 100). A minimax strategy produces as favorable a minimum outcome as possible, no matter what the other might do. The opposite strategy to minimax is to maximize one’s own minimum payoff; this maximin procedure counters an opponent’s strategy of “minimax.” In a two-person constant-sum game with perfect information, the maximin and minimax values of the players are equal (Binmore 2007 ; Hopmann 1996 ). Once an adversary adopts minimax strategies, we cannot do any better by adopting other than a strategy of maximizing our minimum. They are a unique combination of strategies that equalizes all the players’ payoffs. 36 strategic analysis Table 2.5 Minimax: the Battle of the Bismarck Sea Japan North South Allies North 2 2 South 1 3 In Table 2.5 , a strategy of maximizing minimum damage to an adversary is the cor-responding response to their attempt to minimize the maximum loss. It comes from the real-life version of minimax played out in the Battle of the Bismarck Sea in the South Pacific in March 1943 (Carmichael 2005 ). The Japanese navy was convoying rein-forcement troops across the Bismarck Sea from an island called New Britain to a city in nearby New Guinea; the Allies planned to bomb them.
eBook - ePubThe History Of Game Theory, Volume 1
From the Beginnings to 1945
- Mary-Ann Dimand, Robert W Dimand(Authors)
- 1996(Publication Date)
- Taylor & Francis(Publisher)
She then chooses to play the strategy whose minimum payoff is the highest. This is an extremely conservative and pessimistic approach: it assumes that B’s ability to deliver to A her lowest payoff possible, given her choice of strategy, is the paramount element in As choice of strategy. Player A ensures her minimal payoff by taking this approach. A player C taking the minimax approach, on the other hand, looks at the payoffs her opponent D can achieve given each strategy of C.C then chooses to play the strategy which will give D the lowest payoff, if D would always play so as to maximize his payoff subject to C’s strategy. While the maximin approach presumes a player who wishes to guarantee her own minimum payoff, minimax conjectures a player who wants to guarantee her opponents maximum payoff. While the maximin player is conservatively greedy, the minimax player is conservatively aggressive. 1 Aumann (1989, 6–7) stresses the importance of the minimax solution of such games as a ‘vital cornerstone’ for the development of game theory, noting that ‘the most fundamental concepts of the general theory—extensive form, pure strategies, strategic form, randomization, utility theory—were spawned in connection with the minimax theorem’ and that the Cournot—Nash concept of strategic equilibrium in non-cooperative n-person game theory is an outgrowth of minimax. We focus on the contributions made by the eminent French probabilist Borel (1871–1956) in a series of papers from 1921 to 1927, in particular because of the claims of priority which Fréchet (1953) has made for him. We place his work on minimax solution of games of strategy in the context of later papers by von Neumann (1928) and Ville (1938) and of earlier work. BEFORE BOREL The earliest minimax solution of a game was proposed more than two centuries before Borel’s 1921 paper
- Abdelhakim Hammoudi, Nabyla Daidj(Authors)
- 2017(Publication Date)
- Wiley-ISTE(Publisher)
Chapter 1 . As [MAN 96] reminds us, “we speak about uncertainty when, in a given situation, probabilities cannot be calculated. Risk exists when the result is not certain, and when the probability of each possible result is known or can be assessed. Uncertainty arises in a situation when probabilities are unknown. A number of rules have been developed to assist decision makers in making choices from possible attitudes in uncertain conditions, but none are considered to be preferable to others”.This is the case of the Maximin rule that is problematic for acting in situations of uncertainty: the decision maker must determine the worst possible outcome for each type of action and choose the one that has the most desirable worst outcome for his or her firm.2.2.2.2. Generalization: the cautious strategy
Caution is considered to be the state of mind of a player that shows “restraint” in a situation of n strategic interaction. A cautious player first looks to identify the worst situations of his or her opponent and the strategies that lead him or her there. He or she then selects the best situation among the ones that have been identified. In our example, the “best of the worst” for E1 is to obtain 3 by playing strategy A: 3 is therefore called the maximum gain for E1.The following matrix illustrates this notion of caution strategy. Suppose the following gain matrix summarizes the confrontation of two companies E1 and E2: Cautious strategy for E1: (A) Cautious strategy for E2: (A)E2 Min line A B C A A (4, 2) (9, 1) (3, 7) 3 E1 B (5, 4) (2, 3) (4, 0) 2 C (1, 3) (4, 2) (5, 2) 1 Min column 2 1 0 Outcome (A, A) is a cautious strategy. It is the outcome of the game if both players adopt a cautious strategy. However, the applications of the Minimax theorem are limited in the sense that the targeted games are all zero-sum. This is why other solution concepts were researched. This is the case for the concept of Nash equilibrium that is defined for non-descript games (zero-sum and non-zero-sum games). Nash equilibrium is presented in Appendix 2
eBook - PDFEssentials of Game Theory
A Concise Multidisciplinary Introduction
- Kevin Leyton-Brown, Yoav Shoham, Kevin Gebser, Yoav Kaminski(Authors)
- 2022(Publication Date)
- Springer(Publisher)
The maxmin strategy is i ’s best choice when first i must commit to a (possibly mixed) strategy, and then the remaining agents −i observe this strategy (but not i ’s action choice) and choose their own strategies to minimize i ’s expected payoff. In the Battle of the Sexes game (Figure 1.6), the maxmin value for either player is 2/3, and requires the maximizing agent to play a mixed strategy. (Do you see why?) While it may not seem reasonable to assume that the other agents would be solely interested in minimizing i ’s utility, it is the case that i plays a maxmin strategy and the other 16 ESSENTIALS OF GAME THEORY agents play arbitrarily, i will still receive an expected payoff of at least his maxmin value. This means that the maxmin strategy is a sensible choice for a conservative agent who wants to maximize his expected utility without having to make any assumptions about the other agents, such as that they will act rationally according to their own interests, or that they will draw their action choices from some known distributions. The minmax strategy and minmax value play a dual role to their maxmin counterparts. In two-player games the minmax strategy for player i against player −i is a strategy that keeps the maximum payoff of −i at a minimum, and the minmax value of player −i is that minimum. This is useful when we want to consider the amount that one player can punish another without regard for his own payoff. Such punishment can arise in repeated games, as we will see in Section 6. The formal definitions follow. Definition 3.1.2 (Minmax, two-player). In a two-player game, the minmax strategy for player i against player −i is arg min s i max s −i u −i (s i , s −i ), and player −i’s minmax value is min s i max s −i u −i (s i , s −i ). In n-player games with n > 2, defining player i ’s minmax strategy against player j is a bit more complicated. This is because i will not usually be able to guarantee that j achieves minimal payoff by acting unilaterally.
eBook - PDF- Fernando Vega-Redondo(Author)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
They are explained in turn. Heuristically, the maximin is the maximum payoff that player 1 would obtain if 2 could react (optimally) to every strategy on her (i.e., 1’s) part by minimizing 1’s payoff. In a sense, this value corresponds to the payoff that 1 should expect if she were extremely pessimistic over 2’s ability to anticipate her own actions. Formally, it is equal to v 1 ≡ max σ 1 ∈ 1 min σ 2 ∈ 2 σ 1 A σ 2 , (2.9) where we are allowing both players to rely on mixed strategies. Symmetrically, the minimax is defined to embody an analogous “pessimistic” interpretation for player 2, i.e., v 2 ≡ min σ 2 ∈ 2 max σ 1 ∈ 1 σ 1 A σ 2 . (2.10) The fundamental result for bilateral zero-sum games is contained in the following theorem. Zero-sum bilateral games 47 Theorem 2.1 (von Neumann, 1928): Let G be a bilateral and finite zero-sum game. Then (i) There exists some v ∗ ∈ R such that v 1 = v 2 = v ∗ . (ii) For every Nash equilibrium ( σ ∗ 1 , σ ∗ 2 ) , σ ∗ 1 A σ ∗ 2 = v ∗ . Proof: We prove first that v 2 ≥ v 1 . Given any σ 1 ∈ 1 and σ 2 ∈ 2 , we obviously have min σ 2 ∈ 2 σ 1 A σ 2 ≤ σ 1 A σ 2 . (2.11) Then, applying the operator max σ 1 ∈ 1 to both sides of the preceding in-equality, it follows that v 1 = max σ 1 ∈ 1 min σ 2 ∈ 2 σ 1 A σ 2 ≤ max σ 1 ∈ 1 σ 1 A σ 2 (2.12) for any given σ 2 ∈ 2 . Therefore, applying now the operator min σ 2 ∈ 2 to both sides of (2.12), we obtain v 1 ≤ min σ 2 ∈ 2 max σ 1 ∈ 1 σ 1 A σ 2 = v 2 , which proves the desired inequality v 2 ≥ v 1 . We now show that v 1 ≥ v 2 . 23 Let ( σ ∗ 1 , σ ∗ 2 ) be a Nash equilibrium of G (an equilibrium always exists, by Theorem 2.2 below – recall Subsection 2.2.3). From the definition of Nash equilibrium we know σ ∗ 1 A σ ∗ 2 ≥ σ 1 A σ ∗ 2 , ∀ σ 1 ∈ 1 (2.13) σ ∗ 1 A σ ∗ 2 ≤ σ ∗ 1 A σ 2 , ∀ σ 2 ∈ 2 . (2.14) On the other hand, v 1 = max σ 1 ∈ 1 min σ 2 ∈ 2 σ 1 A σ 2 ≥ min σ 2 ∈ 2 σ ∗ 1 A σ 2 . Because, by (2.14), min σ 2 ∈ 2 σ ∗ 1 A σ 2 = σ ∗ 1 A σ ∗ 2 , it follows that v 1 ≥ σ ∗ 1 A σ ∗ 2 .
- Richard Quandt, Dusan Triska(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
PART SIX Strategies and General Equilibrium 20 Maximin vs. Nash Equilibrium: Theoretical Results and Empirical Evidence · Manfred J. Holler and Viggo H~st 1. Introduction Von Neumann and Morgenstern (1944) introduced the maximin solution into the theory of games where solution is plausibly a set of rules for each participant which tell him how to behave in every social situation which may conceivably arise (p. 31). It is understood that the rules ofrational behavior must provide definitely for the possibility of irrational behavior on the part of the others (p. 32). Accordingly, a solution contains the statement of how much the participant under consideration can get if he behaves 'rationally.' This 'can get' is, of course, presumed to be minimum; he may get more if the others make mistakes (p. 33). This contrasts with the widely-held view of contemporary game theory that the solution of a noncooperative game has to be a Nash equilibrium (van Damme, 1987, p. 3). Needless to say, the two positions assume different qualities for the decision-makers: Nash equilibrium presupposes that all agents act ratio-nally and are expected to act rationally while maximin allows for irrational behavior by fellow agents. As a consequence, the question of which of the • The authors gratefully acknowledge the conunents of Peter Skott on an earlier version of this paper 246 Holler and H~t two concepts is the right one cannot be decided on theoretical grounds alone. In this paper, we present the results of an empirical study which, however, builds strongly on the theoretical properties of the two concepts and their relations to one another. More specifically, the empirical study makes use of the fact that for all 2 x 2 matrix games the payoff values of the mixed-strategy Nash equilibrium are equal to the maximin payoffs if the maximin solution implies mixed strategies.
eBook - ePubGame Theory in the Social Sciences
A Reader-friendly Guide
- Luca Lambertini(Author)
- 2011(Publication Date)
- Routledge(Publisher)
Suppose we are looking at a two-player constant-sum game (of course, there may be more than two players, but this is the easiest way of making ourselves acquainted with the first solution concept we are meeting in the book). Each of them receives the following instructions: ‘Given that this is a strictly competitive game, you have to choose the best strategy for yourself, in order to maximize the size of your slice of the pie you see in front of you, being aware that your rival is going to behave in a completely analogous way.’ Or, equivalently: ‘Given that this is a strictly competitive game, you have to choose the best strategy for yourself, in order to maximize the size of your slice of the pie you see in front of you, being aware that your rival is going to choose his/her strategy so as to minimize the size of your slice.’ That is, given the constant-sum nature of the game, each player perceives the rival's behaviour as not only non-cooperative but indeed aggressive. Hence, each player's Maximin Strategy is chosen as a sort of insurance against the harmful consequences of the rival's attitude. A bit more formally, the Maximin Strategy for player 1 guarantees to the latter the attainment of the maximum among the minimum payoffs generated by all admissible strategies available to 1 in the game at hand, given that player 2 is behaving in such a way so as to minimize 1’s maximum payoff (whereby one may adopt the alternative label, minimax). That is, player 1 has to solve the following problem: while 2 is solving this: This, of course, is what 1 thinks that 2 is trying to do, while from the standpoint of 2 the problem is in fact to find the strategy that solves the problem max s 2 min s 1 π 2 (s 1, s 2), being convinced that 1 is trying to solve min s 1 max s 2 π 2 (s 1, s 2). Intuitively, a constant-sum game looks pretty much like a hall of mirrors
- Thomas J. Webster(Author)
- 2018(Publication Date)
- Routledge(Publisher)
What he found was that a rational player will adopt a strategy that simultaneously attempts to maximize gains and minimize losses. When all players adopt this strategy, the maximum of the minimum (maximin) payoffs for all players will be equal. Later, Professor John Nash attempted to predict a player’s strategy in any game, not just zero-sum games. Nash argued that a rational player will always adopt a strategy that is the best response to a rival’s strategy. It was subsequently demonstrated for zero-sum games that the maximin and Nash equilibrium strategies are one and the same. Minimax theorem In a zero-sum, noncooperative game, a player will attempt to minimize a rival’s maximum (minimax) payoff, while maximizing his or her own minimum (maximin) payoff. The minimax and the maximin payoffs of all players will be equal. To illustrate the rationale underlying the minimax theorem, consider the following situation. Two young children, Andrew and Adam, accompany their mother, Rosette, on a trip to a shopping mall. At the mall, Andrew asks his mother to buy him a candy bar. Rosette knows that if she buys a candy bar for Andrew, Adam will want a candy bar as well. Since Rosette is concerned about her children eating too many sweets, she comes up with the following solution. She tells Andrew F IGURE 6.6 Final Iterated Payoff Matrix for the Sink-the-Battleship Game MIXING PURE STRATEGIES 137 that he can have the candy bar, which we will assume is uniform throughout, but must share it with his brother. She also tells Andrew that he can divide the candy bar in any way he wants, but that Adam gets to choose the first piece. Andrew accepts his mother’s offer and divides the candy bar exactly in half. Although Andrew knows nothing about game theory, he has behaved precisely the way the minimax theorem would have predicted. He minimized the maximum size of the piece that Adam gets, while at the same time maximizing the minimum size of his own piece.
- Berç Rustem, Melendres Howe(Authors)
- 2009(Publication Date)
- Princeton University Press(Publisher)
The best decision is computed given that the worst case is determined using all the scenarios. An example of this approach, with two rival macroeconomic models, is discussed in Becker et al. (1986). We introduce the minimax strategy by considering the pooling of rival objective functions. Let q denote pooling weights, with q [ E m sce 1 ; q [ R m sce j q $ 0; k1; ql ¼ 1 n o where 1 [ R m sce denotes the vector with every component unity. Consider the optimal decision problem formulated as the constrained optimization of the pooled objective functions subject to nonlinear constraints min x kq; f ðxÞl j gðxÞ ¼ 0; hðxÞ # 0 f g ð1:1Þ where q [ E m sce 1 , x [ R n , f : R n ! R m sce , g : R n ! R e , h : R n ! R i and f, g and h are twice continuously differentiable functions. In (1.1) each element of f, denoted by f j , represents a rival objective function corresponding to the jth rival model or scenario. The restrictions gðxÞ ¼ 0 and hðxÞ # 0 are the equality and inequality constraints imposed on the overall decision problem. In the above formulation, the vector q is fixed and defines the pooling weights. Also, generally, the number of scenarios or models is much less than the number of decision variables, that is, n q m sce . One reason for this is that policy opti- mization is essentially dynamic in nature and the total number of decision variables is the product of the decision variables for one time period and the number of time periods. Further motivation for (1.1) is discussed in Rustem (1987). Among possible choices of pooling weights, the robust pooling corresponds to the strategy that is invariant to whichever rival scenario, or model, actually turns out to represent the system. The pooling that corresponds to the robust policy is given by the solution of the minimax problem min x max q kf ðxÞ; ql j gðxÞ ¼ 0; hðxÞ # 0; q [ E m sce 1 n o ð1:2Þ where the worst-case rival model scenario is computed simultaneously with the minimization over x. It is shown in Lemma 3.1 below that the solution of (1.2) has a robust character. Whichever rival model turns out to represent the actual system, the optimal (minimax) strategy ensures that the objective
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