{"id":226872,"date":"2025-06-05T16:19:44","date_gmt":"2025-06-05T16:19:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=226872"},"modified":"2025-06-05T16:19:46","modified_gmt":"2025-06-05T16:19:46","slug":"zurich-and-munich","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/05\/zurich-and-munich\/","title":{"rendered":"Zurich and Munich."},"content":{"rendered":"\n

Zurich and Munich. Both are considering adding an additional daily train to this route. The payoff matrix shows the payoffs for each railway, where Swiss Rail values are in italics and EuroRail values are in bold. Assume that both companies have complete knowledge of the other’s payoff matrix. | | Add train | Add train | |———-|———–|———–| | Add train|
1,500 | | Do not add train |
4,000 | | Do not add train |
2,000 | | Do not add train |
3,000 | Select the answer that best describes the strategies in this game. A. Both companies’ dominant strategy is to add the train. B. EuroRail’s dominant strategy is to add the train, whereas Swiss Rail does not have a dominant strategy. C. Neither company has a dominant strategy. D. EuroRail’s dominant strategy is to not add the train, whereas Swiss Rail’s dominant strategy is to add the train. Does a Nash equilibrium exist in this game? A. Yes, it exists in the lower left quadrant. B. No, it does not exist. C. Yes, it exists in the upper right quadrant. D. Yes, it exists in the upper left quadrant.
EuroRail and Swiss Rail are hypothetical railways that have a duopoly on the route that connects the cities of Zurich and Munich. Both are considering adding an additional daily train to this route. The payoff matrix shows the payoffs for each railway, where Swiss Rail values are in italics and EuroRail values are in bold. Assume that both companies have complete knowledge of the other’s payoff matrix. | | Add train | Add train | |———-|———–|———–| | Add train|
1,500 | | Do not add train |
4,000 | | Do not add train |
2,000 | | Do not add train |
3,000 | Select the answer that best describes the strategies in this game. A. Both companies’ dominant strategy is to add the train. B. EuroRail’s dominant strategy is to add the train, whereas Swiss Rail does not have a dominant strategy. C. Neither company has a dominant strategy. D. EuroRail’s dominant strategy is to not add the train, whereas Swiss Rail’s dominant strategy is to add the train. Does a Nash equilibrium exist in this game? A. Yes, it exists in the lower left quadrant. B. No, it does not exist. C. Yes, it exists in the upper right quadrant. D. Yes, it exists in the upper left quadrant.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To solve this game theory problem, we\u2019ll first reconstruct the payoff matrix<\/strong>, identify dominant strategies<\/strong>, and analyze for Nash equilibrium<\/strong>. Then, we\u2019ll provide a comprehensive explanation of the strategic interactions involved.<\/p>\n\n\n\n

\n\n\n\n

1. Payoff Matrix Setup<\/strong><\/h3>\n\n\n\n

From your description, here is the assumed structure<\/strong> of the payoff matrix<\/strong> for Swiss Rail (SR) and EuroRail (ER), formatted as (Swiss Rail, EuroRail):<\/p>\n\n\n\n

<\/th>EuroRail: Add Train<\/th>EuroRail: Do Not Add Train<\/th><\/tr><\/thead>
Swiss: Add<\/strong><\/td>(1500, 1500<\/strong>)<\/td>(4000, 1000<\/strong>)<\/td><\/tr>
Swiss: Do Not Add<\/strong><\/td>(2000, 4000<\/strong>)<\/td>(3000, 3000<\/strong>)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Note: Italics are replaced with normal text, and we represent the matrix as (Swiss, EuroRail). This aligns with the description and creates a clear format for analysis.<\/p>\n\n\n\n

\n\n\n\n

2. Dominant Strategies<\/strong><\/h2>\n\n\n\n

A dominant strategy<\/strong> is one that gives a player a better payoff no matter what the other player does.<\/p>\n\n\n\n

Let\u2019s examine each player:<\/p>\n\n\n\n

\n\n\n\n

Swiss Rail (SR) Strategy<\/strong><\/h3>\n\n\n\n

Compare SR\u2019s payoffs in both ER scenarios:<\/p>\n\n\n\n

    \n
  • If ER Adds a Train:<\/strong>\n
      \n
    • SR Adds: 1500<\/li>\n\n\n\n
    • SR Doesn\u2019t Add: 2000 \u2192 Better<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • If ER Doesn\u2019t Add:<\/strong>\n
        \n
      • SR Adds: 4000<\/li>\n\n\n\n
      • SR Doesn\u2019t Add: 3000 \u2192 Better<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

        So:<\/p>\n\n\n\n

          \n
        • When ER adds \u2192 SR prefers not<\/strong> to add.<\/li>\n\n\n\n
        • When ER doesn\u2019t add \u2192 SR prefers to add<\/strong>.<\/li>\n<\/ul>\n\n\n\n

          \ud83e\udc7a Swiss Rail has no dominant strategy<\/strong>, since its best choice depends on what EuroRail does.<\/p>\n\n\n\n

          \n\n\n\n

          EuroRail (ER) Strategy<\/strong><\/h3>\n\n\n\n

          Compare ER\u2019s payoffs in both SR scenarios:<\/p>\n\n\n\n

            \n
          • If SR Adds a Train:<\/strong>\n
              \n
            • ER Adds: 1500<\/li>\n\n\n\n
            • ER Doesn\u2019t Add: 1000 \u2192 Better to Add<\/strong><\/li>\n<\/ul>\n<\/li>\n\n\n\n
            • If SR Doesn\u2019t Add:<\/strong>\n
                \n
              • ER Adds: 4000<\/li>\n\n\n\n
              • ER Doesn\u2019t Add: 3000 \u2192 Better to Add<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

                \ud83e\udc7a EuroRail always prefers to Add a Train<\/strong>, regardless of Swiss Rail\u2019s choice.<\/p>\n\n\n\n

                \ud83e\udc7a So, EuroRail has a dominant strategy: Add a Train.<\/strong><\/p>\n\n\n\n

                \n\n\n\n

                Answer to Strategy Question<\/strong><\/h3>\n\n\n\n
                \n

                Correct Answer: B. EuroRail’s dominant strategy is to add the train, whereas Swiss Rail does not have a dominant strategy.<\/strong><\/p>\n<\/blockquote>\n\n\n\n

                \n\n\n\n

                3. Nash Equilibrium<\/strong><\/h2>\n\n\n\n

                A Nash Equilibrium<\/strong> is a strategy profile where no player has an incentive to deviate unilaterally.<\/p>\n\n\n\n

                Let\u2019s check each combination for a Nash equilibrium.<\/p>\n\n\n\n

                \n\n\n\n

                (Add, Add)<\/strong> = (1500, 1500)<\/h3>\n\n\n\n
                  \n
                • SR: If switches to \u201cDon\u2019t Add\u201d \u2192 gets 2000 \u2192 better off<\/li>\n\n\n\n
                • ER: Already choosing dominant strategy<\/li>\n<\/ul>\n\n\n\n

                  \ud83e\udc7a Not a Nash Equilibrium<\/strong>, because SR would switch.<\/p>\n\n\n\n

                  \n\n\n\n

                  (Add, Don\u2019t Add)<\/strong> = (4000, 1000)<\/h3>\n\n\n\n
                    \n
                  • SR: Best choice if ER doesn\u2019t add<\/li>\n\n\n\n
                  • ER: But ER can get 3000 by switching to Add \u2192 would switch<\/li>\n<\/ul>\n\n\n\n

                    \ud83e\udc7a Not a Nash Equilibrium<\/strong><\/p>\n\n\n\n

                    \n\n\n\n

                    (Don\u2019t Add, Add)<\/strong> = (2000, 4000)<\/h3>\n\n\n\n
                      \n
                    • SR: Better off than 1500<\/li>\n\n\n\n
                    • ER: Best possible response (dominant strategy)<\/li>\n\n\n\n
                    • SR: Could switch to Add and get 1500 \u2192 worse off<\/li>\n\n\n\n
                    • ER: Won\u2019t deviate (4000 is best)<\/li>\n<\/ul>\n\n\n\n

                      \ud83e\udc7a No one wants to deviate \u2192 this is a Nash Equilibrium<\/strong><\/p>\n\n\n\n

                      \u2705 Yes, Nash equilibrium exists in lower left quadrant<\/strong><\/p>\n\n\n\n

                      \n\n\n\n

                      (Don\u2019t Add, Don\u2019t Add)<\/strong> = (3000, 3000)<\/h3>\n\n\n\n
                        \n
                      • SR: Could get 4000 by switching \u2192 would deviate<\/li>\n\n\n\n
                      • ER: Could get 4000 by switching \u2192 would deviate<\/li>\n<\/ul>\n\n\n\n

                        \ud83e\udc7a Not a Nash Equilibrium<\/strong><\/p>\n\n\n\n

                        \n\n\n\n

                        Answer to Nash Equilibrium Question<\/strong><\/h3>\n\n\n\n
                        \n

                        Correct Answer: A. Yes, it exists in the lower left quadrant.<\/strong><\/p>\n<\/blockquote>\n\n\n\n

                        \n\n\n\n

                        Explanation<\/strong><\/h2>\n\n\n\n

                        Let\u2019s now elaborate with a detailed explanation, bringing in key economic and game theory concepts.<\/p>\n\n\n\n

                        \n\n\n\n

                        A. Background: Oligopoly and Strategic Decisions<\/strong><\/h3>\n\n\n\n

                        The market described is a duopoly<\/strong>, meaning two firms (Swiss Rail and EuroRail) dominate the train route between Zurich and Munich. In such markets, strategic decisions by one firm directly affect the other.<\/p>\n\n\n\n

                        Here, both firms are deciding whether or not to add a new train<\/strong>. Each firm’s profits depend not only on its own decision but also on the decision made by the other company. This interdependence makes the situation ripe for game theoretic analysis<\/strong>.<\/p>\n\n\n\n

                        \n\n\n\n

                        B. Understanding Payoffs<\/strong><\/h3>\n\n\n\n

                        Each firm’s payoffs are affected by competition:<\/p>\n\n\n\n

                          \n
                        • When both add a train, competition intensifies. Result: both earn less (1500 each).<\/li>\n\n\n\n
                        • When only one adds, the firm that adds gains a first-mover or competitive advantage<\/strong> (4000), and the other suffers (1000 or 2000).<\/li>\n\n\n\n
                        • When neither adds, both maintain current profitability (3000 each) \u2014 moderate but stable outcome<\/strong>.<\/li>\n<\/ul>\n\n\n\n

                          The payoffs reflect classic tension in oligopoly:<\/p>\n\n\n\n

                            \n
                          • Aggressive strategy (Add Train)<\/strong>: Could lead to competitive war.<\/li>\n\n\n\n
                          • Passive strategy (Don\u2019t Add)<\/strong>: Risk being undercut.<\/li>\n<\/ul>\n\n\n\n
                            \n\n\n\n

                            C. Dominant Strategy Explained<\/strong><\/h3>\n\n\n\n

                            A dominant strategy<\/strong> offers the best payoff regardless of what the opponent does. We find that:<\/p>\n\n\n\n

                              \n
                            • EuroRail has a dominant strategy<\/strong>: Always do better by adding a train<\/strong>.\n
                                \n
                              • This reflects an aggressive strategy<\/strong> typical of firms with lower opportunity costs or better market reach.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
                              • Swiss Rail lacks a dominant strategy<\/strong>: Its best choice depends on EuroRail\u2019s move.\n
                                  \n
                                • If EuroRail adds, Swiss is better off not adding.<\/li>\n\n\n\n
                                • If EuroRail doesn\u2019t add, Swiss should add.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

                                  This illustrates strategic dependence<\/strong>: Swiss Rail must anticipate EuroRail\u2019s move.<\/p>\n\n\n\n

                                  \n\n\n\n

                                  D. Nash Equilibrium: Stability in Strategy<\/strong><\/h3>\n\n\n\n

                                  A Nash Equilibrium<\/strong> is a pair of strategies where neither player benefits by deviating unilaterally.<\/p>\n\n\n\n

                                  Let\u2019s focus on the (Don\u2019t Add, Add)<\/strong> quadrant:<\/p>\n\n\n\n

                                    \n
                                  • EuroRail<\/strong> adds the train \u2014 its dominant strategy.<\/li>\n\n\n\n
                                  • Swiss Rail<\/strong>, knowing ER adds, chooses not to add since 2000 > 1500.<\/li>\n\n\n\n
                                  • If either deviates:\n
                                      \n
                                    • ER would still choose to Add.<\/li>\n\n\n\n
                                    • SR would lose by deviating.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

                                      \ud83e\udc7a Neither firm benefits by changing its decision \u2192 stable outcome<\/strong>.<\/p>\n\n\n\n

                                      Thus, (Don\u2019t Add, Add)<\/strong> is the only Nash Equilibrium<\/strong> in this game. It\u2019s also asymmetric<\/strong> \u2014 one firm (EuroRail) is aggressive; the other (Swiss Rail) is cautious.<\/p>\n\n\n\n

                                      \n\n\n\n

                                      E. Real-World Implications<\/strong><\/h3>\n\n\n\n

                                      This scenario mirrors real-world oligopolies like:<\/p>\n\n\n\n

                                        \n
                                      • Airline route decisions<\/li>\n\n\n\n
                                      • Tech product launches<\/li>\n\n\n\n
                                      • Gas station pricing<\/li>\n<\/ul>\n\n\n\n

                                        In each, firms must anticipate rivals’ behavior and act strategically. Often, firms with dominant strategies (e.g., lower costs, higher capacity) force others to accommodate.<\/p>\n\n\n\n

                                        In this case:<\/p>\n\n\n\n

                                          \n
                                        • EuroRail\u2019s aggressive strategy forces Swiss Rail to be reactive<\/strong>.<\/li>\n\n\n\n
                                        • If both were aggressive (Add\/Add), both suffer \u2014 the \u201cprisoner\u2019s dilemma\u201d of market competition.<\/li>\n<\/ul>\n\n\n\n

                                          This explains why Nash Equilibrium does not always align with the socially optimal outcome<\/strong> (e.g., (Do Not Add, Do Not Add) where both get 3000). Firms act in self-interest<\/strong>, not collective best interest.<\/p>\n\n\n\n

                                          \n\n\n\n

                                          F. Final Summary<\/strong><\/h3>\n\n\n\n
                                          Aspect<\/th>Answer<\/th><\/tr><\/thead>
                                          Dominant Strategies<\/strong><\/td>EuroRail has a dominant strategy to Add; Swiss Rail does not.<\/strong><\/td><\/tr>
                                          Correct Multiple Choice<\/strong><\/td>B<\/strong><\/td><\/tr>
                                          Nash Equilibrium<\/strong><\/td>Exists in Lower Left Quadrant<\/strong> \u2014 Swiss: Don\u2019t Add, EuroRail: Add<\/td><\/tr>
                                          Correct Multiple Choice<\/strong><\/td>A<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
                                          \n\n\n\n
                                          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

                                          Zurich and Munich. Both are considering adding an additional daily train to this route. The payoff matrix shows the payoffs for each railway, where Swiss Rail values are in italics and EuroRail values are in bold. Assume that both companies have complete knowledge of the other’s payoff matrix. | | Add train | Add train […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-226872","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/226872","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=226872"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/226872\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=226872"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=226872"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=226872"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}