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I recently have come across a business problem which could be convereted into a game problem as follows:

Imagine an infinitely repeated game between two players in which the firts player (leader) first makes a decision and then based on that decision the second player (follower) decides. According to the logic of rationality, in each stage (repeatition) the second player should optimise her own plan. But if, in some stages of game instead of optimising her plan, the second player makes some other decisions, a very high cost will be incurred to the first player. This incurred cost could change the way that the first player makes decision to avoid the risk caused by the second player plays differently, and consequently it could increase the profit of the second player in long term.

In brief, one of the players makes a decision that impose cost to himself and his opponent but it could change the decision making of the opponent and in long term offers higher profit for him. Has this type of game addressed in any game theory books?

I personally, categorised it as an strategy in playing the game. A strategy could contain some threats to change the behavior of other players. However, the the threat must be credibile. When one of the palyer's decision in punishing (threatening) the other player is different with Nash Equilibria, the threat may not be considered as a credibel one. But I feel this could also be improtant to deviate from the Nash equilibria (not take the action of "best response") to chaneg the other player's action and gain some profit afterwards. I beleive in some markets that there is monopoly, this kind of strategies exist. for example, the person (brand) that has the dominant share of market reduces the price (and his profit) to make the business non-profitable for himself and the newcomers, until the newcomers get broke and then they buy their share and increase the price again. Please help me with this problem. Any hint could be useful.

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I do not think an infinitely repeated game is necessary here. If player 1's actions are to take a $risk$ or play it $safe$ and player 2's actions are to $optimize$ or $punish$, the next step is to theorize a probability for actions. For example, player 1 might take a $risk$ with probability $p_1$ if player 2 has a small market cap, but they might $risk$ with probability $p_2$ if player 2 has a smaller market cap. Thus they would play it $safe$ with probability $1-p_1$ or $1-p_2$. This models the situation you mention where companies take a loss to try and make their competitor go broke. This can be modeled with a Bayesian Nash Equilibrium. But first, you must also hypothesize some preference ordering. Under the situation with player 2 having a high market cap, player 1 might prefer $safe|optimize>safe|punish>risk|optimize>risk|punish$ and player 2 might prefer $safe|optimize>risk|punish>risk|optimize>safe|punish$. However, in a world where player 2 has a small market cap, player 1 might prefer $risk|optimize>safe|optimize>risk|punish>safe|punish$ and player 2 might prefer $safe|optimize>safe|punish>risk|punish>risk|optimize$.

Thus, there are two games under uncertainty:

High Market Cap       Low Market Cap
     Optimize Punish       Optimize Punish
Risk 3,3      4,2     Risk 1,4      3,3
Safe 1,2      2,4     Safe 2,1      4,2

Thus, $\Gamma_1$ where player 2 has a high market cap is a Harmony by Assurance game and $\Gamma_2$ where player 2 has a low market cap is a Prisoner's Dilemma by Chicken game. As above, there probabilities of player 1 using risk in both games, $p_1$ and $p2$, and there are probabilities of player 1 being safe in both games, $1-p_1$ and $1-p_2$. There's also probability or player 2 optimizing in both games, $q_1$ and $q_2$, and of punishing in both games, $1-q_1$ and $1-q_2$. There's also a probability of the high market cap, $p(\Gamma_1)$, and of the low market cap, $p(\Gamma_2)=1-p(\Gamma_1)$. There's also conditional probabilities of these outcomes. That is all up to you to calculate. You may also decide to try other preference orderings that are not strictly dominating, like where player 2 has is indifferent between $risk|optimize$ and $safe|punish$. Overall, I believe that exploring the repeated nature of this potential game is less informative than exploring the uncertain nature of this game.

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    $\begingroup$ Thank you very much for understanding the question and your comprehensive answer. It shows your oustanding knwoledge in game theory. My first thought about this problem was formulating it as a problem with mixed strategy in which each player randomises his/her actions to maximise his/her total benefit. The randimisation of actions somehow I feel is equivalent to the idea of probability distribution that you mentioned in the context of Baysian game and considering uncertainty. I am not quite convinced about the uncertain nature of problem. My reason is all the actions and utilities are known. $\endgroup$ – farshid evazabadian Nov 14 '18 at 22:23
  • $\begingroup$ You did not indicate that the utilities are known. That is why I went with a Bayesian game. If the utilities are known, then the repeated nature of the game might be worth exploring. $\endgroup$ – Jay Schyler Raadt Nov 15 '18 at 14:08
  • $\begingroup$ In case that all the information is known to both parties, is there any game in the literature that describe this type of relationship? one of the parties by hurting himself try to hurt the opponent since he knows he will be stronger and can resist more and then when the opponent is gone he can become dominant? $\endgroup$ – farshid evazabadian Nov 16 '18 at 12:43
  • $\begingroup$ In the case that they can withstand the loss, it would be like the PD by chicken game above. $\endgroup$ – Jay Schyler Raadt Nov 16 '18 at 13:44
  • $\begingroup$ Both assurance and Chicken are Coordination-type games $\endgroup$ – Jay Schyler Raadt Nov 16 '18 at 19:02

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