2048 strategy4/5/2023 ![]() Such problems are all around us, and MDPs find many applications in economics, finance, and artificial intelligence. Markov Decision Processes ( MDPs) are a mathematical framework for modeling and solving problems in which we need to make a sequence of related decisions in the presence of uncertainty. The (research quality) code behind this article is open source. We will however be able to find an optimal policy for the shortened 4x4 game to the 64 tile, and fortunately we’ll see that optimal play on the 3x3 boards looks qualitatively similar, in an admittedly hand-wavy way, to some successful strategies for the full game. This makes it infeasible to construct a complete optimal policy for the full game, at least with the methods used here. Ideally we’d be able to find an optimal policy for the full game on the 4x4 board to the 2048 tile, but as we saw in the previous post, the number of possible board configurations is very large. The 2x2 games are qualitatively quite different to the 4x4 games, but they’ll still be useful to introduce the key ideas. It turns out that the 2x2 game to the 32 tile is very hard to win - even when playing optimally, the player only wins about 8% of the time, which probably does not make for a very fun game. ![]() In this post, we’ll see how to construct a policy that is optimal, in the sense that it maximizes the player’s chances of reaching the target 32 tile. The ‘strategy’ the player follows is defined by a table, called a policy, that tells it which direction it should swipe in every possible board configuration. The random seed determines the random sequence of tiles that the game adds to the board. For example, here is an optimal player for the 2x2 game to the 32 tile: ![]() In this post, we’ll use a mathematical framework called a Markov Decision Process to find provably optimal strategies for 2048 when played on the 2x2 and 3x3 boards, and also on the 4x4 board up to the 64 tile. ![]() Most identified collaborations are with businesses, whereas initiatives addressing consumers are largely missing although considered critical for the transition toward Circular Economy.So far in this series on the mathematics of 2048, we’ve used Markov chains to learn that it takes at least 938.8 moves on average to win, and we’ve explored the number of possible board configurations in the game using combinatorics and then exhaustive enumeration. Most reported activities are oriented toward the main product and packaging, focusing on end-of-life management and sourcing strategies, and to a lesser extent on circular product design and business model strategies. Our results show that Circular Economy has started to be integrated into the corporate sustainability agenda. We focus on (i) companies’ uptake of Circular Economy, (ii) the relationship between Circular Economy and sustainability and (iii) the Circular Economy practices presented. To contribute to filling this gap, we perform a systematic review of 46 corporate sustainability reports in the Fast-Moving Consumer Goods sector aiming to explore how companies incorporate the Circular Economy concept in their sustainability agenda. Despite the increasing interest of business and academic research toward Circular Economy, the investigation of its uptake by industry remains limited. ![]()
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