ABSTRACT Usually, real game problems encountered in our daily lives are so complicated that the existing methods are no longer sufficient to deal with them. This motivates us to investigate several kinds of differential game problems, which have not been considered or solved yet, including a pursuit-evasion game with n pursuers and one evader, a problem of guarding a territory with two guarders and two invaders, and a payoff-switching differential game. In this thesis, firstly the geometric method is used to consider the pursuit-evasion game with n pursuers and one evader. Two criteria used to find the solutions of the game in some cases are given. It will be shown that the one-on-one pursuit-evasion game is a special case of this game. Secondly, the problem of guarding a territory with two guarders and two invaders is considered both qualitatively and quantitatively. The investigation of this problem reveals a variety of situations never occurring in the case with one guarder and one invader. An interesting thing found in this investigation is that some invader may play the role as a pursuer for achieving a more favorable payoff in some cases. This will make the problem more complicated and more difficult to be solved. The payoff-switching differential game, first proposed by us, is a kind of differential game with incomplete information. The main difference between this problem and traditional differential games is that in a payoff-switching differential game, any one player at any time may have several choices of payoffs for the future. The optimality in such a problem becomes questionable. Some reasoning mechanisms based on different methods will be provided to determine a reasoning strategy for some player in a payoff-switching differential game. A practical payoff-switching differential game problem, i.e., the guarding three territories with one guarder against one invader, is presented to illustrate the situations of such a game problem. Many computer simulations of this example are given to show the performances of different reasoning strategies. The proposition of the payoff-switching differential game is an important breakthrough in dealing with some kinds of differential games with incomplete information.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / One of the greatest reasons to use robotics rather than human beings is to avoid hazardous situations such as activities related to search, surveillance and rescue. The pursuit-evasion problem is a fundamental theoretical base to apply robotics on these cases. This dissertation presents an approach to solve the pursuit-evasion problem with no previous knowledge of the map, which must be simply connected, using multi-robots systems with limited sensing. The approach is based on the random walk, since it is a mathematical formalization probabilistically complete, considering plane and obstacle free environments that shall be treated discretely through a regular occupation grid. This dissertation also presents a variation of this approach, though it considers random walk probabilities, to enhance the previous approach, decreasing the amount of iterations needed to solve the problem. In order to validate what is proposed, a discrete multi-robot simulation environment was developed. Finally, the results obtained on the tests that were performed and possible future works that could improve this approach are discussed. / Uma das principais motivações do uso de sistemas robóticos em detrimento de seres humanos é evitar situações de risco, como as encontradas em atividades de busca, vigilância e resgate. O problema de perseguição-evasão é uma base teórica fundamental para a aplicação da robótica nestes casos. Esta dissertação apresenta uma abordagem para solução do problema de perseguição-evasão sem um conhecimento a priori do mapa, que deverá ser simplesmente conectado, através da coordenação de múltiplos robôs com visão limitada. A abordagem aqui proposta é baseada na random walk, por esta ser uma formalização matemática probabilisticamente completa, sendo contemplados ambientes planos e sem obstáculos, que serão tratados discretamente por meio de uma grade de ocupação regular. Ainda nesta dissertação, foi proposta uma variação dessa abordagem, porém com a ponderação de probabilidades da random walk, com o objetivo de aprimorar a anterior, diminuindo número de iterações necessárias para solução do problema. Para a validação da abordagem proposta, foi desenvolvido um ambiente de simulações para abordagens discretas de múltiplos robôs. Finalmente, são discutidos os resultados obtidos nos testes realizados e propostos trabalhos futuros para melhoria desta abordagem.

One group of games studied within game theory are grid-based pursuit evasion games of imperfect information. A pursuit evasion game is in essence a game where there exists a set of pursuers which have as their objective to capture a set of evaders. This thesis aims to develop a formalisation of this type of games as well as describing and integrating vital game theoretical concepts such as order of knowledge into this game. With the developed formalism at hand, the concept of knowledge-based strategies is then introduced, which is essential when searching for the way to play the game most efficiently. The formalisation of the game is then followed by a simulation, measuring the performance of some older and some newly developed knowledge-based strategies. The thesis concludes that the formalisation is applicable on a more general class of pursuit evasion games and enables a wider study of the game. The simulation results indicate that knowledge-based strategies of higher order do not always perform better compared to simpler strategies of lower order of knowledge. Furthermore, strategies which allow for communication between agents are found to be superior to communication-less strategies. / En typ av spel som studeras inom spelteori är rutnätsbaserade jakt-flykt-spel med ofullständig information. Ett jakt-flykt-spel går ut på att det existerar en samling jagande aktörer som försöker fånga en samling flyende aktörer. Denna uppsats söker utveckla en formalism för denna typ av spel såväl som att beskriva och integrera ett antal nyckelkoncept inom spelteori såsom kunskapsordning. Med hjälp av den utvecklade formalismen, framställs så kallade kunskapsbaserade strategier, vilka är av fundamental vikt i sökandet efter sätt att spela spelet på det effektivaste sättet. Kapitlet om formalismen följs sedan av simuleringar där några äldre och några nyare kunskapsbaserade strategier prövas. Slutsatsen dras att den nya formalismen kan vara applicerbar på en bredare samling jakt-flykt-spel än den initialt påtänkta. Vidare underlättar formalismen en generalisering till andra sätt att beskriva spel. Simulationsresultaten indikerar att kunskapsbaserade strategier av högre ordning inte alltid presterar bättre än enklare strategier av lägre ordning. Till yttermera visso visar sig kommunikationslösa strategier vara underlägsna strategier som tillåter kommunikation. / Kandidatexjobb i elektroteknik 2022, KTH, Stockholm

Imagine for a moment we are living in the distant future where autonomous robots are patrollingthe streets as police officers. Two such robots are chasing a robber through the city streets. Fearingthe thief might listen in to any potential transmission, both robots remain radio silent and are thuslimited to a strictly visual pursuit. Since the robots cannot see the robber the entire time, they haveto deduce the potential location of the robber. What would the best strategy be for these robots toachieve their objective? This bachelor's thesis investigated the above example by creating strategies through reinforcementlearning. The thesis also investigated the performance of the players when they have differentabilities of deduction. This was tested by creating a suitable game and corresponding reinforcementlearning algorithm and running the simulations for different degrees of knowledge. The study provedthat reinforcement learning is a viable method for strategy construction, reaching nearly guaranteedvictory for cases when the agent knows everything about the environment and a slightly lower winratio when there is uncertainty introduced. The implementation yielded only a small gain in win ratiowhen the agents could deduce even more about each other. / Föreställ dig för ett ögonblick att vi lever i en avlägsen framtid där autonoma robotar patrullerar pågatorna som poliser. Två sådana robotar jagar en rånare genom stadens gator. Eftersom de är räddaför att tjuven kan lyssna på alla möjliga sändningar, förblir båda robotarna radiotysta och är därförbegränsade till en strikt visuell strävan. Eftersom robotarna inte kan se rånaren hela tiden, måste dehärleda den potentiella platsen för rånaren. Vilken skulle den bästa strategin vara för dessa robotarför att uppnå sitt mål? Denna kandidatuppsats undersökte ovanstående exempel genomskapa strategier genomförstärkningsinlärning. Avhandlingen undersökte också spelarnas prestationer när de har olikaavdragsförmåga. Detta testades genom att skapa ett lämpligt spel och motsvarandeförstärkningsinlärningsalgoritm och köra simuleringarna för olika kunskapsgrader. Studien visade attförstärkningsinlärning är en användbar metod för strategikonstruktion, och når nästan garanteradseger i fall då agenten vet allt om miljön och en något lägre vinstkvot när det finns osäkerhet.Implementeringen gav bara en liten vinst i vinstförhållandet när agenterna kunde härleda ännu merom varandra. / Kandidatexjobb i elektroteknik 2023, KTH, Stockholm

Strategies in games have since long been of interestto humans, mainly to beat our friends in games such as Chessor Monopoly, but also to model real world scenarios. Thesestrategies are often difficult to find, even more so if the playerslack important information about the current state of the game.Pursuit-Evasion games are a type of games that can be usedto model police chasing criminals, autonomous car collisionavoidance systems and many other scenarios. It is therefore ofinterest to find effective strategies in these scenarios.This bachelor thesis project examined Pursuit-Evasion gamesof imperfect information on grids where a number of pursuerswork together to capture a number of evaders whose locationsare unknown. A set of knowledge-based strategies, one of theminspired by the Knowledge Based Subset Construction, wereexplored and analyzed. The strategies were compared againsteach other and against both an optimal strategy where thepursuers always were aware of the evaders whereabouts anda reference strategy where the pursuers moved randomly.The constructed strategies proved to be efficient in comparisonto the reference and in cases even close to the optimal strategyin efficiency. / Strategier i spel har sedan länge varit avintresse för oss människor, framförallt för att vinna mot vårakompisar i spel som Schack eller Monopol, men också för attmodellera verkliga scenarion. Dessa strategier är ofta svåra attlista ut, och ännu svårare då spelarna saknar viktig informationom spelets nuvarande läge. Pursuit-Evasion är en klass av spelsom kan användas för att modellera polisjakter eller kollisionsundvikandesystem i autonoma bilar för att nämna några. Detligger därför i vårt intresse att finna effektiva strategier i dessascenarion.Detta kandidatsexamensarbete studerade Pursuit-Evasion spelav imperfekt information på rutnät där ett antal så kallade pursuerssamarbetade för att fånga ett antal så kallade evaders varspostioner var okända. En kunskaps-representation utformadesoch en mängd kunskaps-baserade strategier, en inspirerad avmetoden Knowledge Based Subset Construction, utforskades ochtestades. De olika strategierna jämfördes mot varandra och motbåde en optimal strategi då pursuers hade all kunskap om varalla evaders befann sig och en referensstrategi då pursuers rördesig slumpmässigt.De utformade strategierna visade sig vara effektiva i jämförelsemed referensen och i vissa fall till och med nära den optimalastrategin i effektivitet. / Kandidatexjobb i elektroteknik 2021, KTH, Stockholm

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